In this paper, we characterize monotonically countably paracompact (or monoton-ically countably metacompact) spaces by semi-continuous maps with values intosome ordered topological vector spaces or some topological vector lattices. Theseextend earlier results for real-valued semi-continuous functions.
Throughout this paper, all spaces are assumed to be Hausdorff topological spaces. Let R be the set of allreal numbers, N the set of all natural numbers, κ an infinite cardinal, and ω the first infinite cardinal. (Topo-logical) vector spaces always mean real (topological) vector spaces. C. Good, R. Knight and I. Stares [9] andC. Pan [22] introduced a monotone version of countably paracompact spaces, called monotonically count-ably paracompact spaces and monotonically cp-spaces, respectively, and it is proved in [9, Proposition 14]that both these notions are equivalent.
For two sequences (An)n∈ω and (Bn)n∈ω of subsets of a space, it is written that (An) � (Bn) if An ⊂ Bn
for each n ∈ ω.
Definition 1.1. ([9,22]) A topological space X is said to be monotonically countably metacompact if thereexists an operator U assigning to each decreasing sequence (Dj)j∈ω of closed subsets of X with
⋂j∈ω Dj = ∅,
a sequence U((Dj)) = (U(n, (Dj)))n∈ω of open subsets of X such that
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52 K. Yamazaki / Topology and its Applications 169 (2014) 51–70
(1) Dn ⊂ U(n, (Dj)) for each n ∈ ω;(2)
⋂n∈ω U(n, (Dj)) = ∅;
(3) If (Dj) � (Ej), then U((Dj)) � U((Ej)).
X is said to be monotonically countably paracompact if, in addition,
(2′)⋂
n∈ω U(n, (Dj)) = ∅.
This definition is due to C. Good, R. Knight and I. Stares [9]. See also [22] for an equivalent definitiondue to C. Pan, and [29] for various equivalent conditions of monotonically countably paracompact spacesand monotonically countably metacompact spaces due to C. Ying and C. Good. Note that monotonicallycountably metacompact spaces coincide with β-spaces [9, Theorem 5].
The following two theorems were proved in [8, Theorem 3], [9, Theorem 25] and [26, Theorem 2.4 andCorollary 3.3].
Theorem 1.2. ([8,9,26]) For a topological space X, the following conditions are equivalent:
(1) X is monotonically countably paracompact.(2) There exists an operator Φ assigning to each locally bounded function f : X → R, a locally bounded lower
semi-continuous function Φ(f) : X → R with |f | � Φ(f) such that Φ(f) � Φ(f ′) whenever |f | � |f ′|.(3) There exist operators Φ, Ψ assigning to each upper semi-continuous function f : X → R with f � 0,
a lower semi-continuous function Φ(f) : X → R and an upper semi-continuous function Ψ(f) : X → R
with f � Φ(f) � Ψ(f) such that Φ(f) � Φ(f ′) and Ψ(f) � Ψ(f ′) whenever f � f ′.
Theorem 1.3. ([26]) For a topological space X, the following conditions are equivalent:
(1) X is monotonically countably metacompact.(2) There exists an operator Φ assigning to each locally bounded function f : X → R, a lower semi-
continuous function Φ(f) : X → R with |f | � Φ(f) such that Φ(f) � Φ(f ′) whenever |f | � |f ′|.(3) There exists an operator Φ assigning to each upper semi-continuous function f : X → R with f � 0,
a lower semi-continuous function Φ(f) : X → R with f � Φ(f) such that Φ(f) � Φ(f ′) wheneverf � f ′.
The purpose of this paper is to generalize real-valued functions in Theorems 1.2 and 1.3 to maps withvalues into some ordered topological vector spaces Y . This provides some advantage to the real-valuedcases. Indeed, the range R with the total order can be extended to spaces Y with the partial order. Also,our results do not make use of basic tools depending on Banach space theories or selection principles.
In the rest of this section, let us recall some definitions and terminology from [24], see also [1,2,7]. Fora topological space X and A ⊂ X, intX A and A denote the interior and the closure, respectively, of A
in X.The origin of a vector space is denoted by 0. Let Y be a vector space. For y ∈ Y and A,B ⊂ Y , define
−A = {−a: a ∈ A}, A + B = {a + b: a ∈ A, b ∈ B}, A−B = A + (−B), and y + A = {y} + A. Note thatthe complement of A in Y (= {y ∈ Y : y /∈ A}) is denoted by Y \ A. A subset A ⊂ Y is circled if λA ⊂ A
whenever |λ| � 1. For a map f : X → Y , the map −f : X → Y is defined by (−f)(x) = −f(x) for eachx ∈ X. The map 0 : X → Y is defined by 0(x) = 0 for each x ∈ X.
A partially ordered vector space (Y,�) is said to be an ordered vector space if the following conditionsare satisfied:
K. Yamazaki / Topology and its Applications 169 (2014) 51–70 53
(i) x � y implies x + z � y + z for all x, y, z ∈ Y ,(ii) x � y implies rx � ry for all x, y ∈ Y and all r ∈ R with r � 0.
Sometimes, x � y will be denoted by y � x. Let (Y,�) be an ordered vector space. Then, y ∈ Y is positiveif 0 � y, and the set {y ∈ Y : y � 0}, called the positive cone of Y , is always denoted by Y +. For y1, y2 ∈ Y ,write y1 < y2 if y1 � y2 and y1 �= y2. Subspaces (y1 +Y +)∩ (y2 − Y +) of Y , where y1, y2 ∈ Y with y1 � y2,are called order intervals. A subset A ⊂ Y is said to be order bounded if A ⊂ (y1 + Y +) ∩ (y2 − Y +) forsome y1, y2 ∈ Y . For maps f, g : X → Y we write f � g if f(x) � g(x) for each x ∈ X.
A topological vector space Y is called an ordered topological vector space if Y is an ordered vector spacesuch that the positive cone Y + is closed in Y .
For a topological vector space and an ordered vector space Y , the positive cone Y + is normal if each0-neighborhood U admits a 0-neighborhood V such that (V + Y +) ∩ (V − Y +) ⊂ U .
A vector lattice is an ordered vector space satisfying the following condition:
(iii) Every two-point set {x, y} has a least upper bound x ∨ y and a greatest lower bound x ∧ y.
Let Y be a vector lattice. For each y ∈ Y put |y| = y∨ (−y). A subset A of Y is said to be solid if ‘x ∈ A
and |y| � |x|’ implies y ∈ A. For maps f, g : X → Y , the maps |f | : X → Y and f ∨ g : X → Y (resp.f ∧ g : X → Y ) are defined by |f |(x) = |f(x)| and (f ∨ g)(x) = f(x) ∨ g(x) (resp. (f ∧ g)(x) = f(x) ∧ g(x))for each x ∈ X. For A,B ⊂ Y , define A ∨B = {a ∨ b: a ∈ A, b ∈ B}.
Let Y be a topological vector space and a vector lattice. Then, Y is called locally solid if Y possesses a0-neighborhood base of solid sets. It is known that Y is locally solid if and only if the positive cone Y + isnormal and lattice operations are continuous [24, Chapter V, 7.1]. Also, Y is said to be a topological vectorlattice if it is locally solid. Note that every topological vector lattice is an ordered topological vector space[24, Chapter V, 7.1]. A normed lattice Y is a vector lattice with a norm ‖·‖ such that the following conditionis satisfied:
(iv) |x| � |y| implies ‖x‖ � ‖y‖ for all x, y ∈ Y .
A normed lattice is said to be a Banach lattice if it is a Banach space.Familiar examples of topological vector lattices (in fact, Banach lattices) are Banach spaces lp(κ), where
1 � p � ∞, C∗(Z) and C0(Z). Here, C∗(Z) is the set of all bounded continuous functions s on a topologicalspace Z, C0(Z) is the set of all continuous functions s on a topological space Z such that for each ε > 0 theset {z ∈ Z: |s(z)| � ε} is compact. The linear operations on C∗(Z) and C0(Z) are defined pointwise andthe topology is introduced by the norm ‖s‖ = supz∈Z |s(z)| for each s ∈ C∗(Z) and s ∈ C0(Z). The symbolc0(κ) denotes the space C0(Z) when Z is the discrete space of cardinality κ, and c0 = c0(ω). As usual, fors, t ∈ C∗(Z) (resp. s, t ∈ C0(Z), lp(κ), 1 � p � ∞), s � t means s(z) � t(z) for each z ∈ Z (resp. z ∈ Z,z ∈ κ). Banach spaces Lp([0, 1]), where 1 � p � ∞, are also topological vector lattices (in fact, Banachlattices) with their natural orderings, and, for s, t ∈ Lp([0, 1]), s � t means s(z) � t(z) a.e. on z ∈ [0, 1].
Let X and Y be topological spaces. A set valued-mapping ϕ : X → 2Y is lower semi-continuous ifϕ−1[U ] = {x ∈ X: ϕ(x) ∩ U �= ∅} is open in X for each open set U of Y . For an ordered topologicalvector space Y , J.M. Borwein and M. Théra [5] define a map f : X → Y to be lower semi-continuous if theset-valued mapping ϕ : X → 2Y , defined by ϕ(x) = f(x) − Y + for x ∈ X, is lower semi-continuous, andf is upper semi-continuous if −f is lower semi-continuous (see also [4,23]).
Recall [7] that a real-valued function f : X → R is lower semi-continuous if {x: f(x) > r} is openfor each r ∈ R (namely, for each x ∈ X and each ε > 0 there exists a neighborhood Ox of x such thatf(x′) > f(x) − ε for each x′ ∈ Ox). A real-valued function f : X → R is upper semi-continuous if −f is
54 K. Yamazaki / Topology and its Applications 169 (2014) 51–70
lower semi-continuous. Note that this definition coincides with the above definition of semi-continuous mapswith values into ordered topological vector spaces Y in [5] for Y = R.
Definitions of semi-continuous maps to ordered topological vector spaces are recently formulated asfollows:
Proposition 1.4. (Cf. [4,5,11,23,27].) Let X be a topological space, Y an ordered topological vector space,and f : X → Y a map. Then, the following conditions (1) and (2) are equivalent:
(1) f is lower semi-continuous (resp. upper semi-continuous).(2) For each x ∈ X and each 0-neighborhood V there exists a neighborhood Ox of x such that f(x′) ∈
f(x) + V + Y + (resp. f(x′) ∈ f(x) + V − Y +) for each x′ ∈ Ox.
If we assume further that Y is a topological vector lattice, the following condition (3) is equivalent to (1):
(3) For each x ∈ X and each 0-neighborhood V there exists a neighborhood Ox of x such that f(x)− f(x)∧f(x′) ∈ V (resp. f(x) ∨ f(x′) − f(x) ∈ V ) for each x′ ∈ Ox.
If we assume further that Y is a normed space, the following condition (4) is equivalent to (1):
(4) For each x ∈ X and each ε > 0 there exists a neighborhood Ox of x such that ‖f(x)− f(x)∧ f(x′)‖ < ε
(resp. ‖f(x) ∨ f(x′) − f(x)‖ < ε) for each x′ ∈ Ox.
In case of Y = C0(Z), where Z is a topological space, the following condition (5) is equivalent to (1):
(5) For each x ∈ X and each ε > 0 there exists a neighborhood Ox of x such that f(x′)(z) > f(x)(z) − ε
(resp. f(x′)(z) < f(x)(z) + ε) for each z ∈ Z and each x′ ∈ Ox.
Proof. The equivalence (1) ⇔ (2) was proved in [4,5,23] (see also [27, Proposition 2.1]), while (1) ⇔ (3) in[27, Proposition 2.2], and (1) ⇔ (4) in [27, Corollary 2.3 (1)]. (5) is the definition of semi-continuous mapsf : X → C0(Z) in the sense of [11, Definition 2.1] (see also [27, Corollary 2.3 (2)]). �
By (1) ⇔ (2) of Proposition 1.4, we have that for a topological space X and an ordered topological vectorspace Y such that the positive cone Y + is normal, a map f : X → Y is continuous if and only if f is lowerand upper semi-continuous (cf. [23,27]).
2. Basic lemmas
Let Y be an ordered topological vector space and e ∈ Y +. Then, e is called an interior point of Y + ife ∈ intY (Y +); e is an internal point of Y + if for each y ∈ Y there exists λ0 > 0 such that e + λy ∈ Y + foreach λ with 0 � λ � λ0; e is an order unit if for each y ∈ Y there exists λ > 0 such that y � λe [2]. Recall[2, Lemmas 1.7 and 2.5, and Theorem 2.8] the following:
Proposition 2.1. (See [2].) For an ordered topological vector space Y and e ∈ Y +, the implications (1) ⇔(2) ⇒ (3) ⇔ (4) hold.
(1) e is an interior point of Y +.(2) (−e + Y +) ∩ (e− Y +) is a 0-neighborhood.
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(3) e is an internal point of Y +.(4) e is an order unit.
If we assume further that Y is completely metrizable, (3) is equivalent to (1).
For an ordered topological vector space Y and e ∈ Y , we call e a positive interior point if e is an interiorpoint of Y + and e > 0. If Y is a non-trivial (that is, Y �= {0}) ordered topological vector space and e is aninterior point of Y +, then e is a positive interior point. Indeed, assume on the contrary that e = 0, thentake a 0-neighborhood U such that U ⊂ Y + and U = −U . Since Y is non-trivial, take u ∈ U \ {0}, whichprovides that u,−u ∈ Y +, hence we have u = 0, a contradiction.
Of course, R and C∗(Z), where Z (�= ∅) is a topological space, have interior points. On the other hand,neither c0 nor lp (1 � p < ∞) has interior points (or equivalently, order units). Also, C0(Z) does not haveinterior points for each non-compact locally compact space Z. To see this, assume on the contrary that C0(Z),where Z is non-compact locally compact, has an interior point e > 0. By (1) ⇔ (2) of Proposition 2.1, takeε > 0 such that B(0; ε) ⊂ (−e+ (C0(Z))+) ∩ (e− (C0(Z))+), where B(0; ε) is the ε-ball of 0. Since the setA = {z ∈ Z: e(z) � ε/2} is compact, Z �= A. Hence, take z0 ∈ Z \ A and a neighborhood O of z0 in Z
such that O is compact and O ∩ A = ∅. Take a continuous function f : Z → [0, ε/2] such that f(z0) = ε/2and f−1(0, ε/2] ⊂ O. Then, f ∈ C0(Z) and f ∈ B(0; ε) \ (−e + (C0(Z))+) ∩ (e − (C0(Z))+), because O iscompact, ‖f‖ = ε/2 and f(z0) = ε/2 > e(z0). A contradiction.
We first give a basic proposition on semi-continuous maps f : X → Y to ordered topological vectorspaces Y . Obviously, a real-valued function f : X → R is lower semi-continuous if and only if {x: f(x) � r}is closed for each r ∈ R; notice that a simple analogue of this fact for maps f : X → Y to ordered topologicalvector spaces Y is not true as Example 2.3 shows.
Proposition 2.2. Let X be a topological space, Y an ordered topological vector space and f : X → Y a map.Then, implications (1) ⇔ (2) ⇒ (3) hold.
(1) f is lower semi-continuous (resp. upper semi-continuous).(2) f−1(y + V + Y +) (resp. f−1(y + V − Y +)) is open for each open 0-neighborhood V and each y ∈ Y .(3) f−1(y − Y +) (resp. f−1(y + Y +)) is closed for each y ∈ Y .
Proof. (1) ⇒ (2): Assume (1). Let V be an open 0-neighborhood, y ∈ Y and x ∈ f−1(y + V + Y +). Sincey+V +Y + is open in Y , take a 0-neighborhood W such that f(x) +W ⊂ y +V +Y +. By (1), there existsa neighborhood Ox of x such that f(Ox) ⊂ f(x) + W + Y +. Hence, for each x′ ∈ Ox,
f(x′) ∈ f(x) + W + Y + ⊂
(y + V + Y +) + Y + = y + V + Y +,
therefore, Ox ⊂ f−1(y + V + Y +). Thus, (2) holds.(2) ⇒ (1): For x ∈ X and a 0-neighborhood V , set Ox = f−1(f(x) + intY V + Y +).(1) ⇒ (3): Assume (1). Let y ∈ Y and x /∈ f−1(y − Y +). Since Y + is closed, y − Y + is also closed, and
hence, there exists a 0-neighborhood V such that (f(x)+V )∩(y−Y +) = ∅. By (1), take a neighborhood Ox
of x such that f(Ox) ⊂ f(x) + V + Y +. Then, we have Ox ∩ f−1(y − Y +) = ∅. Indeed, for each x′ ∈ Ox,there exist v ∈ V and s ∈ Y + such that f(x′) = f(x) + v + s. Hence, f(x′)− s = f(x) + v /∈ y − Y +, whichimplies that f(x′) /∈ y−Y +. This shows that Ox∩f−1(y−Y +) = ∅. Thus, f−1(y−Y +) is closed, therefore,(3) holds.
The respective part follows, because f : X → Y is upper semi-continuous if and only if −f : X → Y islower semi-continuous. �
56 K. Yamazaki / Topology and its Applications 169 (2014) 51–70
Example 2.3. In Proposition 2.2, the implication (3) ⇒ (1) does not necessarily hold. To see this, letX = {0, 1/n: n ∈ N} ⊂ R, and Y = C∗(N), that is, the set of all bounded real-valued functions on N withthe sup-norm topology and the usual order relation. Consider the map f : X → Y defined by f(x)(i) = 0if x = 1/n and i < n; f(x)(i) = −1 if x = 1/n and i � n; f(0) = 0 (= (0, 0, . . .)).
To show that f satisfies (3) of Proposition 2.2, let y = (y(n))n∈N ∈ Y and consider the following twocases.
Case 1. f−1(y − Y +) is infinite. Namely, {x ∈ X: f(x) � y} is infinite. Hence, for every n ∈ N, {x ∈ X:f(x)(n) � y(n)} is infinite, and this provides that y(n) � 0. Thus, by the definition of f , y = (y(n)) � 0 �f(x) for each x ∈ X. It follows that f(X) ⊂ y − Y +, therefore, f−1(y − Y +) = X is closed.
Case 2. f−1(y − Y +) is finite. Then, it is clear that f−1(y − Y +) is closed.
To see that f does not satisfy (1) of Proposition 2.2, let V be the (1/2)-ball of 0. We have f(O) �⊂f(0) + V + Y + = V + Y + for each neighborhood O of 0 in X. Hence, f is not lower semi-continuous.
Let X be a topological space and Y an ordered topological vector space. Then, we call a map f : X → Y
locally upper bounded if for each x ∈ X and each 0-neighborhood V , there exist a neighborhood Ox of xand n ∈ N such that f(Ox) ⊂ nV − Y +. A map f : X → Y is locally lower bounded if and only if −f islocally upper bounded. It is obvious that f : X → Y is locally lower bounded if and only if for each x ∈ X
and each 0-neighborhood V , there exist a neighborhood Ox of x and n ∈ N such that f(Ox) ⊂ nV + Y +.
Proposition 2.4. Let X be a topological space, Y an ordered topological vector space and f : X → Y a map.Then, the following conditions (1) and (2) are equivalent:
(1) f : X → Y is locally upper bounded (resp. locally lower bounded).(2) {int(f−1(nV − Y +)): n ∈ N} (resp. {int(f−1(nV + Y +)): n ∈ N}) is an increasing open cover of X
for each circled 0-neighborhood V .
If we assume further that Y has an interior point e of Y +, then the following conditions (3), (4) and (5)are mutually equivalent to (1).
(3) For each x ∈ X, there exist a neighborhood Ox of x and n ∈ N such that f(Ox) ⊂ ne − Y + (resp.f(Ox) ⊂ −ne + Y +).
(4) {int(f−1(ne− Y +)): n ∈ N} (resp. {int(f−1(−ne + Y +)): n ∈ N}) is an increasing open cover of X.(5) There exists an upper semi-continuous (resp. a lower semi-continuous) map Φ(f) : X → Y such that
f � Φ(f) (resp. Φ(f) � f).
Proof. (1) ⇔ (2): Easy.Assume, in the following, further that Y has an interior point e of Y +.(1) ⇒ (3): Assume (1), and let x ∈ X. Then, (−e + Y +) ∩ (e− Y +) is a 0-neighborhood from (1) ⇔ (2)
of Proposition 2.1. By (1), there exist a neighborhood Ox of x and n ∈ N such that f(Ox) ⊂ n((−e+Y +)∩(e− Y +)) − Y + = ne− Y +. Thus, (3) holds.
(3) ⇒ (4): Easy.(4) ⇒ (5): Assume (4). For each x ∈ X, set n(x) = min{n ∈ N: x ∈ int(f−1(ne − Y +))}. Define
a map Φ(f) : X → Y by Φ(f)(x) = n(x) · e for each x ∈ X. Since f(x) ∈ n(x) · e − Y +, we havef(x) � n(x) · e = Φ(f)(x) for each x ∈ X, and hence, f � Φ(f). To see Φ(f) is upper semi-continuous, letx ∈ X. Set Ox = int(f−1(n(x) · e − Y +)). Then, for each x′ ∈ Ox, we have n(x′) � n(x) by the definition
K. Yamazaki / Topology and its Applications 169 (2014) 51–70 57
of n(x′), and hence Φ(f)(x′) = n(x′) · e � n(x) · e = Φ(f)(x). This shows that Φ(f)(Ox) ⊂ Φ(f)(x) − Y +,which provides that Φ(f) is upper semi-continuous. Thus, (5) holds.
(5) ⇒ (1): Assume (5), and let Φ(f) : X → Y be an upper semi-continuous map such that f � Φ(f).Let x ∈ X and V be a 0-neighborhood. Since (−e + Y +) ∩ (e − Y +) is a 0-neighborhood by (1) ⇔ (2) ofProposition 2.1, there exists a neighborhood Ox of x such that
Φ(f)(Ox) ⊂ Φ(f)(x) +(−e + Y +) ∩
(e− Y +)− Y +.
Take n ∈ N with Φ(f)(x) ∈ n((−e+Y +)∩(e−Y +)). For each x′ ∈ Ox, there exist v, v′ ∈ (−e+Y +)∩(e−Y +)such that Φ(f)(x′) � Φ(f)(x)+v and Φ(f)(x) = nv′, this shows that Φ(f)(x′) � nv′+v � ne+e = (n+1)e.Hence, we have Φ(f)(Ox) ⊂ (n + 1)e − Y +. On the other hand, take m ∈ N with e ∈ mV . It follows thatΦ(f)(Ox) ⊂ (n+1)e−Y + ⊂ (n+1)mV −Y +. Thus, f(Ox) ⊂ (n+1)mV −Y +, this shows that f is locallyupper bounded. Therefore, (1) holds.
For the respective part, use the fact that f : X → Y is locally lower bounded (resp. lower semi-continuous)if and only if −f is locally upper bounded (resp. upper semi-continuous). �Lemma 2.5. Let Y be an ordered topological vector space with an interior point of Y +. Then, there existsan operator Φ assigning to a locally upper bounded map f : X → Y , an upper semi-continuous mapΦ(f) : X → Y with f � Φ(f) such that Φ(f) � Φ(f ′) whenever f � f ′.
Proof. Let e be an interior point of Y + in Y , f : X → Y a locally upper bounded map, and x ∈ X. By(1) ⇔ (4) of Proposition 2.4, we can set n(x)(f) = min{n ∈ N: x ∈ int(f−1(ne − Y +))}. Consider themap Φ(f) : X → Y defined by Φ(f)(x) = n(x)(f) · e for each x ∈ X. By the similar proof of (4) ⇒ (5) ofProposition 2.4, Φ(f) is upper semi-continuous with f � Φ(f). For a locally upper bounded map f ′ : X → Y
with f � f ′, it follows from (f ′)−1(ne − Y +) ⊂ f−1(ne − Y +) for each n ∈ N that n(x)(f) � n(x)(f ′) foreach x ∈ X. Hence, we have Φ(f)(x) = n(x)(f) · e � n(x)(f ′) · e = Φ(f ′)(x) for each x ∈ X, therefore,Φ(f) � Φ(f ′). �
For a topological space X, a real-valued function f : X → R is said to be locally bounded if for each x ∈ X
there exist a neighborhood Ox of x and n ∈ N such that −n < f(x′) < n for each x′ ∈ Ox. It is natural todefine as follows: f : X → R is locally upper bounded if for each x ∈ X there exist a neighborhood Ox of xand n ∈ N such that f(x′) < n for each x′ ∈ Ox; f : X → R is locally lower bounded if −f is locally upperbounded. Of course, f : X → R is locally bounded if and only if f is locally upper bounded and locallylower bounded.
We now call a map f : X → Y locally bounded from a topological space X to an ordered topologicalvector space Y if for each x ∈ X and each 0-neighborhood V , there exist a neighborhood Ox of x and n ∈ N
such that f(Ox) ⊂ nV . In the following proposition, we give basic facts of locally bounded maps f : X → Y
to ordered topological vector spaces Y , extending those of locally bounded real-valued functions f : X → R.The proof is easy and omitted.
Proposition 2.6. Let X be a topological space, Y an ordered topological vector space and f : X → Y a map.Then, the following are valid.
(1) f is locally bounded if and only if {int(f−1(nV )): n ∈ N} is an increasing open cover of X for eachcircled 0-neighborhood V .
(2) If f is locally bounded, then f is locally upper bounded and locally lower bounded. The converse isalso true if we assume further that the positive cone Y + is normal. In particular, for topological vectorlattices Y , the converse is true.
58 K. Yamazaki / Topology and its Applications 169 (2014) 51–70
Lemma 2.7. For a topological space X and a topological vector lattice Y , the following are valid.
(1) If maps f, g : X → Y are lower semi-continuous (resp. upper semi-continuous), then f ∨ g and f ∧ g
are lower semi-continuous (resp. upper semi-continuous).(2) If a map f : X → Y is locally upper bounded (resp. locally lower bounded) and g : X → Y is a map,
then f ∧ g (resp. f ∨ g) is locally upper bounded (resp. locally lower bounded).(3) If Y + has an interior point in Y and maps f, g : X → Y are locally upper bounded (resp. locally lower
bounded), then f ∨ g (resp. f ∧ g) is locally upper bounded (resp. locally lower bounded).
Proof. (1) Let f, g : X → Y be lower semi-continuous maps, x ∈ X and V a 0-neighborhood. SinceY is a topological vector lattice, the sup-operation ∨ : Y × Y → Y ; (y1, y2) �→ y1 ∨ y2, is continuous.Hence, there exists a 0-neighborhood U such that (f(x) + U) ∨ (g(x) + U) ⊂ f(x) ∨ g(x) + V . Since f, g
are lower semi-continuous, there exists a neighborhood Ox of x such that f(Ox) ⊂ f(x) + U + Y + andg(Ox) ⊂ g(x) + U + Y +. Then, for each x′ ∈ Ox, there exist u, u′ ∈ U such that f(x′) � f(x) + u andg(x′) � g(x) + u′, and hence,
(f ∨ g)(x′) = f
(x′) ∨ g
(x′) �
(f(x) + u
)∨(g(x) + u′) ∈ f(x) ∨ g(x) + V = (f ∨ g)(x) + V.
This shows that (f ∨ g)(Ox) ⊂ (f ∨ g)(x) + V + Y +, which implies that f ∨ g is lower semi-continuous.Lower semi-continuity of f ∧ g is similar to show. For a respective part, use the facts that −(f ∧ g) =
(−f) ∨ (−g), and that f is lower semi-continuous if and only if −f is upper semi-continuous.(2) Easy.(3) Let e be an interior point of Y + in Y , f, g : X → Y locally upper bounded maps, and x ∈ X. By
(1) ⇔ (3) of Proposition 2.4, take a neighborhood Ox of x and n ∈ N such that f(Ox) ⊂ ne − Y + andg(Ox) ⊂ ne− Y +. Then, (f ∨ g)(Ox) ⊂ ne− Y +, this shows that f ∨ g is locally upper bounded.
For the respective part, use the fact that f is locally lower bounded if and only if −f is locally upperbounded. �Corollary 2.8. Let X be a topological space, and Y an ordered topological vector space with an interior point eof Y +. Assume that the positive cone Y + is normal. Then, for a map f : X → Y , the following conditions(1) and (2) are equivalent:
(1) f is locally bounded.(2) For each x ∈ X, there exist a neighborhood Ox of x and n ∈ N such that f(Ox) ⊂ (−ne+Y +)∩(ne−Y +).
In particular, when Y is a topological vector lattice, (3) is equivalent to (1).
(3) There exists an upper semi-continuous map Φ(f) : X → Y such that |f | � Φ(f).
Proof. (1) ⇒ (2): Assume (1), and let x ∈ X. By (1) ⇔ (2) of Proposition 2.1, (−e + Y +) ∩ (e− Y +) is a0-neighborhood. By (1), there exist a neighborhood Ox of x and n ∈ N such that
f(Ox) ⊂ n((−e + Y +) ∩
(e− Y +)) =
(−ne + Y +) ∩
(ne− Y +).
Hence, (2) holds.(2) ⇒ (1): Assume (2). It follows from (1) ⇔ (3) of Proposition 2.4 that f is locally upper bounded and
locally lower bounded. By (2) of Proposition 2.6, f is locally bounded. Hence, (1) holds.Assume, in the following, further that Y is a topological vector lattice.
K. Yamazaki / Topology and its Applications 169 (2014) 51–70 59
(1) ⇒ (3): Assume (1). Then, by (2) of Proposition 2.6, we have f,−f are locally upper bounded. By(1) ⇔ (5) of Proposition 2.4, there exist upper semi-continuous maps g1, g2 : X → Y such that f � g1 and−f � g2. Hence, |f | = f ∨ (−f) � g1 ∨ g2, and g1 ∨ g2 is upper semi-continuous by (1) of Lemma 2.7. Thus,(3) holds.
(3) ⇒ (1): Assume (3), and let Φ(f) : X → Y be an upper semi-continuous map with |f | � Φ(f). Sincef � |f | � Φ(f), it follows from (1) ⇔ (5) of Proposition 2.4 that f is locally upper bounded. Similarly,−f � |f | � Φ(f) provides that −f is locally upper bounded, namely, f is locally lower bounded. Hence,from (2) of Proposition 2.6 we obtain that f is locally bounded. So, (1) holds. �3. Main theorems
Extending Theorem 1.2 (that is, [8,9,26]), we have:
Theorem 3.1. Let X be a topological space and Y an ordered topological vector space with a positive interiorpoint. Then, the following conditions (1), (2) and (3) are equivalent:
(1) X is monotonically countably paracompact.(2) There exists an operator Φ assigning to each locally upper bounded map f : X → Y , a locally upper
bounded lower semi-continuous map Φ(f) : X → Y with f � Φ(f) such that Φ(f) � Φ(f ′) wheneverf � f ′.
(3) There exist operators Φ, Ψ assigning to each upper semi-continuous map f : X → Y , a lower semi-continuous map Φ(f) : X → Y and an upper semi-continuous map Ψ(f) : X → Y with f � Φ(f) � Ψ(f)such that Φ(f) � Φ(f ′) and Ψ(f) � Ψ(f ′) whenever f � f ′.
If we assume further that Y is a topological vector lattice, (4) is equivalent to (1):
(4) There exists an operator Φ assigning to each locally bounded map f : X → Y , a locally bounded lowersemi-continuous map Φ(f) : X → Y with |f | � Φ(f) such that Φ(f) � Φ(f ′) whenever |f | � |f ′|.
Proof. Let e be a positive interior point of Y .(1) ⇒ (2): Assume (1). Let U be an operator satisfying (1), (2′) and (3) in Definition 1.1. We may assume
U also satisfies the following:
U(n + 1, (Dj)
)⊂ U
(n, (Dj)
)for each n ∈ ω; (3.1)
U(0, (Dj)
)= X. (3.2)
Let f : X → Y be a locally upper bounded map. For each n ∈ ω, set Fn(f) = f−1(Y \ (ne− Y +)). Then,{Fn(f): n ∈ ω} is a decreasing sequence of closed subsets of X with
⋂n∈ω Fn(f) = ∅ because of (1) ⇔ (4)
of Proposition 2.4. Hence, U((Fj(f))) = (U(n, (Fj(f))))n∈ω is a sequence of open subsets of X such that:
Fn(f) ⊂ U(n,
(Fj(f)
))for each n ∈ ω; (3.3)
⋂
n∈ω
U(n,
(Fj(f)
))= ∅; (3.4)
If(Fj(f)
)�
(Fj
(f ′)), then U
((Fj(f)
))� U
((Fj
(f ′))); (3.5)
U(n + 1,
(Fj(f)
))⊂ U
(n,
(Fj(f)
))for each n ∈ ω; and (3.6)
U(0,(Fj(f)
))= X. (3.7)
60 K. Yamazaki / Topology and its Applications 169 (2014) 51–70
By (3.4), we set n(x)(f) = min{n ∈ ω: x /∈ U(n, (Fj(f)))}, and define a map Φ(f) : X → Y by Φ(f)(x) =n(x)(f) · e for each x ∈ X. It follows from (3.7) that
n(x)(f) � 1 for each x ∈ X. (3.8)
To show f � Φ(f), let x ∈ X. Then, by the definition of n(x)(f) and (3.3),
x /∈ U(n(x)(f),
(Fj(f)
))⊃ Fn(x)(f)(f) = f−1
(Y \
(n(x)(f) · e− Y +
)),
which provides that f(x) ∈ n(x)(f) · e − Y +. Hence, f(x) � n(x)(f) · e = Φ(f)(x), and thus, we havef � Φ(f).
To show that Φ(f) : X → Y is locally upper bounded, let x ∈ X. By (3.4), there exists m ∈ ω such thatx /∈ U(m, (Fj(f))). Consider the neighborhood Ox = X \ U(m, (Fj(f))) of x. For each x′ ∈ Ox, it followsfrom the definition of n(x′)(f) that n(x′)(f) � m. Hence, Φ(f)(x′) = n(x′)(f) · e � me. This shows thatΦ(f)(Ox) ⊂ me− Y +. By (1) ⇔ (3) of Proposition 2.4, Φ(f) is locally upper bounded.
To show that Φ(f) : X → Y is lower semi-continuous, let x ∈ X. Since n(x)(f) � 1 by (3.8), considerthe neighborhood Ox = U(n(x)(f) − 1, (Fj(f))) of x. For each x′ ∈ Ox, it follows from the definition ofn(x′)(f) that n(x′)(f) > n(x)(f) − 1, hence Φ(f)(x′) = n(x′)(f) · e � n(x)(f) · e = Φ(f)(x). Therefore,Φ(f)(Ox) ⊂ Φ(f)(x) + Y +, this provides that Φ(f) : X → Y is lower semi-continuous.
Finally, let f ′ : X → Y be a locally upper bounded map with f � f ′. Then,
f−1(Y \(ne− Y +)) ⊂
(f ′)−1(
Y \(ne− Y +)),
and hence, Fn(f) ⊂ Fn(f ′) for each n ∈ ω. Therefore, we have U((Fj(f))) � U((Fj(f ′))). To see Φ(f) �Φ(f ′), let x ∈ X. By the definition of n(x)(f ′), we have
x /∈ U(n(x)
(f ′),
(Fj
(f ′))) ⊃ U
(n(x)
(f ′),
(Fj(f)
)).
Also, by the definition of n(x)(f), we have n(x)(f) � n(x)(f ′). Hence, Φ(f)(x) = n(x)(f) ·e � n(x)(f ′) ·e =Φ(f ′)(x), so we obtain Φ(f) � Φ(f ′). Thus, (2) holds.
(2) ⇒ (3): Let Φ be an operator assigning to each locally upper bounded map g : X → Y , a locally upperbounded and lower semi-continuous map Φ(g) : X → Y with g � Φ(g) such that
Φ(g) � Φ(g′)
whenever g � g′. (3.9)
By Lemma 2.5, there exists an operator ψ assigning to a locally upper bounded map g : X → Y , to anupper semi-continuous map ψ(g) : X → Y with g � ψ(g) such that
ψ(g) � ψ(g′)
whenever g � g′. (3.10)
Let f : X → Y be an upper semi-continuous map. By (1) ⇔ (5) of Proposition 2.4, f is locallyupper bounded. Since Φ(f) is locally upper bounded, define Ψ(f) = ψ(Φ(f)). Now, we have f � Φ(f) �ψ(Φ(f)) = Ψ(f). Also, for an upper semi-continuous map f ′ : X → Y with f � f ′, it follows from (3.9) thatΦ(f) � Φ(f ′), hence, by (3.10), we obtain Ψ(f) = ψ(Φ(f)) � ψ(Φ(f ′)) = Ψ(f ′). Thus, Φ, Ψ are requiredones in (3).
(3) ⇒ (1): Let Φ, Ψ be operators as in (3). Let (Dj)j∈ω be a decreasing sequence of closed subsets ofX with
⋂j∈ω Dj = ∅. Define a map f((Dj)) : X → Y by f((Dj))(x) = n(x)((Dj)) · e for each x ∈ X,
where n(x)((Dj)) = min{n ∈ ω: x /∈ Dn}. To show f((Dj)) : X → Y is upper semi-continuous, let x ∈ X.Since x /∈ Dn(x)((Dj)), consider the neighborhood Ox = X \ Dn(x)((Dj)) of x. For each x′ ∈ Ox, we have
K. Yamazaki / Topology and its Applications 169 (2014) 51–70 61
n(x′)((Dj)) � n(x)((Dj)), hence, f((Dj))(x′) = n(x′)((Dj)) ·e � n(x)((Dj)) ·e = f((Dj))(x). This shows thatf((Dj))(Ox) ⊂ f((Dj))(x) − Y +. Thus, f((Dj)) : X → Y is upper semi-continuous.
Now, Φ(f((Dj))) : X → Y is lower semi-continuous, Ψ(f((Dj))) : X → Y is upper semi-continuous withf((Dj)) � Φ(f((Dj))) � Ψ(f((Dj))), and
for each n ∈ ω.To see Dn ⊂ U(n, (Dj)) for each n ∈ ω, fix n ∈ ω and let x ∈ Dn. Then,
n(x)((Dj)
)� n + 1. (3.12)
Since Φ(f((Dj))) is lower semi-continuous and (−e/2 + Y +) ∩ (e/2 − Y +) is a 0-neighborhood by (1) ⇔ (2)of Proposition 2.1, there exists a neighborhood Ox of x such that
Φ(f((Dj)))(Ox) ⊂ Φ(f((Dj)))(x) + int Y
((−e/2 + Y +) ∩
(e/2 − Y +)) + Y +.
We prove that Ox ⊂ Φ(f((Dj)))−1(Y \ (ne − Y +)). To do this, let x′ ∈ Ox. Take v ∈ int Y ((−e/2 + Y +) ∩(e/2− Y +)) and s ∈ Y + such that Φ(f((Dj)))(x′) = Φ(f((Dj)))(x) + v + s. Since e > 0, we have −e < −e/2.Hence, −e/2 � v, which implies that −e < v. Therefore, by (3.12),
Φ(f((Dj)))(x′) � Φ(f((Dj)))(x) + v > f((Dj))(x) − e = n(x)
((Dj)
)· e− e � (n + 1)e− e = ne,
which provides that Φ(f((Dj)))(x′) ∈ Y \ (ne − Y +). Thus, Ox ⊂ Φ(f((Dj)))−1(Y \ (ne − Y +)), this showsthat x ∈ U(n, (Dj)).
Next, we show that⋂
n∈ω U(n, (Dj)) = ∅. To see this, let x ∈ X. Since Φ(f((Dj))) � Ψ(f((Dj))) andΨ(f((Dj))) : X → Y is upper semi-continuous, by (3) ⇔ (5) of Proposition 2.4, there exist a neighborhood Ox
of x and n ∈ ω such that Φ(f((Dj)))(Ox) ⊂ ne− Y +. We have Ox ∩Φ(f((Dj)))−1(Y \ (ne− Y +)) = ∅. Thus,x /∈ U(n, (Dj)). Therefore,
⋂n∈ω U(n, (Dj)) = ∅.
Let ((Ej)) be a decreasing sequence of closed subsets of X with the empty intersection and (Dj) � (Ej).Since Dn ⊂ En for each n ∈ ω, for each x ∈ X, it follows from
n(x)((Dj)
)= min{n ∈ ω: x /∈ Dn} � min{n ∈ ω: x /∈ En} = n(x)
((Ej)
)
that
f((Dj))(x) = n(x)((Dj)
)· e � n(x)
((Ej)
)· e = f((Ej))(x).
Hence, it follows from (3.11) that Φ(f((Dj))) � Φ(f((Ej))). So, we have
Φ(f((Dj)))−1(Y \
(ne− Y +)) ⊂ Φ(f((Ej)))
−1(Y \(ne− Y +))
for each n ∈ ω. Thus, U(n, (Dj)) ⊂ U(n, (Ej)) for each n ∈ ω, which implies that U((Dj)) � U((Ej)).Hence, (1) holds.
Assume, in the following, further that Y is a topological vector lattice.
62 K. Yamazaki / Topology and its Applications 169 (2014) 51–70
(2) ⇒ (4): Let φ be an operator assigning to each locally upper bounded map g : X → Y , a locally upperbounded lower semi-continuous map φ(g) : X → Y with g � φ(g) such that
φ(g) � φ(g′)
whenever g � g′. (3.13)
Let f : X → Y be a locally bounded map. Then, maps f,−f : X → Y are locally upper bounded by (2)of Proposition 2.6. Then, |f | = f ∨ (−f) is locally upper bounded by (3) of Lemma 2.7. Set Φ(f) = φ(|f |),then Φ(f) : X → Y is locally upper bounded lower semi-continuous. Also, Φ(f) = φ(|f |) � |f | � 0, whichprovides that Φ(f) is locally lower bounded. Hence, Φ(f) is locally bounded by (2) of Proposition 2.6.
For a locally bounded map f ′ : X → Y with |f | � |f ′|, we have Φ(f) = φ(|f |) � φ(|f ′|) = Φ(f ′) by (3.13).Thus, Φ is a required one in (4).
(4) ⇒ (2): Let φ be an operator assigning to each locally bounded map g : X → Y , a locally boundedlower semi-continuous map φ(g) : X → Y with |g| � φ(g) such that
φ(g) � φ(g′)
whenever |g| �∣∣g′
∣∣. (3.14)
Let f : X → Y be a locally upper bounded map. Since 0 : X → Y is locally upper bounded, f ∨ 0 is locallyupper bounded by (3) of Lemma 2.7. Also, since 0 : X → Y is locally lower bounded, f ∨ 0 is locally lowerbounded by (2) of Lemma 2.7. Hence, f ∨ 0 is locally bounded by (2) of Proposition 2.6. Define Φ(f) =φ(f ∨ 0). Then, Φ(f) is locally bounded lower semi-continuous, and f � f ∨ 0 = |f ∨ 0| � φ(f ∨ 0) = Φ(f).
Let f ′ : X → Y be a locally upper bounded map with f � f ′. Then, |f ∨ 0| = f ∨ 0 � f ′ ∨ 0 = |f ′ ∨ 0|.It follows from (3.14) that Φ(f) = φ(f ∨ 0) � φ(f ′ ∨ 0) = Φ(f ′). Thus, Φ is a required one in (2). �
Extending Theorem 1.3 (that is, [26]), we have:
Theorem 3.2. Let X be a topological space and Y an ordered topological vector space with a positive interiorpoint. Then, the following conditions (1), (2) and (3) are equivalent:
(1) X is monotonically countably metacompact.(2) There exists an operator Φ assigning to each locally upper bounded map f : X → Y , a lower semi-
continuous map Φ(f) : X → Y with f � Φ(f) such that Φ(f) � Φ(f ′) whenever f � f ′.(3) There exists an operator Φ assigning to each upper semi-continuous map f : X → Y , a lower semi-
continuous map Φ(f) : X → Y with f � Φ(f) such that Φ(f) � Φ(f ′) whenever f � f ′.
If we assume further that Y is a topological vector lattice, then (4) is equivalent to (1):
(4) There exists an operator Φ assigning to each locally bounded map f : X → Y , a lower semi-continuousmap Φ(f) : X → Y with |f | � Φ(f) such that Φ(f) � Φ(f ′) whenever |f | � |f ′|.
Proof. The proof is obtained by a modification of that of Theorem 3.1. So, we only show the outline of theproof. Let e be a positive interior point of Y .
(1) ⇒ (2): Assume (1). Let U be an operator satisfying (1), (2) and (3) in Definition 1.1. We may assumethat U also satisfies the following: U(n + 1, (Dj)) ⊂ U(n, (Dj)) for each n ∈ ω; U(0, (Dj)) = X. Letf : X → Y be a locally upper bounded map. For each n ∈ ω, set Fn(f) = f−1(Y \ (ne− Y +)). Then,{Fn(f): n ∈ ω} is a decreasing sequence of closed subsets of X with
⋂n∈ω Fn(f) = ∅. Set n(x)(f) =
min{n ∈ ω: x /∈ U(n, (Fj(f)))}, and define a map Φ(f) : X → Y by Φ(f)(x) = n(x)(f) · e for each x ∈ X.Then, Φ(f) : X → Y is lower semi-continuous such that f � Φ(f). For a locally upper bounded mapf ′ : X → Y with f � f ′, we have Φ(f) � Φ(f ′). Thus, (2) holds.
K. Yamazaki / Topology and its Applications 169 (2014) 51–70 63
(2) ⇒ (3): Let Φ be an operator as in (2). Then, Φ is a required one, because every upper semi-continuousmap f : X → Y is locally upper bounded by (1) ⇔ (5) of Proposition 2.4. Thus, (3) holds.
(3) ⇒ (1): Let Φ be an operator as in (3). Let (Dj)j∈ω be a decreasing sequence of closed subsets of Xwith
⋂j∈ω Dj = ∅. Define a map f((Dj)) : X → Y by f((Dj))(x) = n(x)((Dj)) · e for each x ∈ X, where
n(x)((Dj)) = min{n ∈ ω: x /∈ Dn}. Then, f((Dj)) : X → Y is upper semi-continuous. Set U(n, (Dj)) =int{(Φ(f((Dj))))−1(Y \ (ne − Y +))} for each n ∈ ω. Then, Dn ⊂ U(n, (Dj)) for each n ∈ ω. Now, let usshow that
⋂n∈ω U(n, (Dj)) = ∅. To show this, let x ∈ X. Since [−e, e] is a 0-neighborhood by (1) ⇔ (2)
Proposition 2.1, there exists n ∈ ω such that Φ(f((Dj)))(x) ∈ n[−e, e]. Hence, Φ(f((Dj)))(x) ∈ ne− Y +. Wehave
x /∈ Φ(f((Dj)))−1(Y \
(ne− Y +)) ⊃ U
(n, (Dj)
),
which shows that⋂
n∈ω U(n, (Dj)) = ∅. For a decreasing sequence ((Ej)) of closed subsets of X with theempty intersection with (Dj) � (Ej), we have U((Dj)) � U((Ej)). Thus, (1) holds.
Assume, in the following, further that Y is a topological vector lattice.(2) ⇒ (4): Let φ be an operator assigning to each locally upper bounded map g : X → Y , a lower
semi-continuous map φ(g) : X → Y with g � φ(g) such that φ(g) � φ(g′) whenever g � g′. Let f : X → Y
be a locally bounded map. Then, |f | is locally upper bounded. Set Φ(f) = φ(|f |), and then Φ(f) : X → Y
is lower semi-continuous with |f | � φ(|f |) = Φ(f). For a locally bounded map f ′ : X → Y with |f | � |f ′|,we have Φ(f) = φ(|f |) � φ(|f ′|) = Φ(f ′). Thus, (4) holds.
(4) ⇒ (2): Let φ be an operator assigning to each locally bounded map g : X → Y , a lower semi-continuousmap φ(g) : X → Y with |g| � φ(g) such that φ(g) � φ(g′) whenever |g| � |g′|. Let f : X → Y be alocally upper bounded map. Then, f ∨ 0 : X → Y is locally bounded, and define Φ(f) = φ(f ∨ 0). Then,f � f ∨ 0 = |f ∨ 0| � φ(f ∨ 0) = Φ(f). For a locally bounded map f ′ : X → Y with f � f ′, we haveΦ(f) = φ(f ∨ 0) � φ(f ′ ∨ 0) = Φ(f ′). Thus, (2) holds. �4. Other results
By analogy with Theorems 3.1 and 3.2, we can prove the following Theorems 4.1 and 4.2, which extendsome earlier results.
In Theorem 4.1, the equivalences (1) ⇔ (3) and (1) ⇔ (4) extend (1) ⇔ (6) of [26, Corollary 3.3] (seealso [18, Lemma 7]) and [17, Theorem 10], respectively, where Y = R.
Theorem 4.1. Let X be a topological space and Y an ordered topological vector space with a positive interiorpoint. Then, the following conditions (1), (2) and (3) are equivalent:
(1) X is countably paracompact.(2) For each locally upper bounded map f : X → Y , there exists a locally upper bounded lower semi-
continuous map Φ(f) : X → Y such that f � Φ(f).(3) For each upper semi-continuous map f : X → Y , there exist a lower semi-continuous map Φ(f) : X → Y
and an upper semi-continuous map Ψ(f) : X → Y such that f � Φ(f) � Ψ(f).
If we assume further that Y is a topological vector lattice, then (4) is equivalent to (1):
(4) For each locally bounded map f : X → Y , there exists a locally bounded lower semi-continuous mapΦ(f) : X → Y such that |f | � Φ(f).
In Theorem 4.2, the equivalences (1) ⇔ (3) and (1) ⇔ (4) extend (1) ⇔ (6) of [26, Corollary 3.3] and(1) ⇔ (6) of [26, Theorem 2.4], respectively, where Y = R.
64 K. Yamazaki / Topology and its Applications 169 (2014) 51–70
Theorem 4.2. Let X be a topological space and Y an ordered topological vector space with a positive interiorpoint. Then, the following conditions (1), (2) and (3) are equivalent:
(1) X is countably metacompact.(2) For each locally upper bounded map f : X → Y , there exists a lower semi-continuous map Φ(f) : X → Y
such that f � Φ(f).(3) For each upper semi-continuous map f : X → Y , there exists a lower semi-continuous map Φ(f) : X →
Y such that f � Φ(f).
If we assume further that Y is a topological vector lattice, then (4) is equivalent to (1):
(4) For each locally bounded map f : X → Y , there exists a lower semi-continuous map Φ(f) : X → Y suchthat |f | � Φ(f).
A space is said to be a cb-space [12] if for each locally bounded function f : X → R there exists acontinuous function Φ(f) : X → R such that |f | � Φ(f). Recall [17, Theorem 1] and [19, Theorem 1.2] thatX is a cb-space if and only if every countable increasing open cover is normal. Every cb-space is countablyparacompact, and every normal countably paracompact space is a cb-space. Corresponding to Theorems 4.1and 4.2, we have the following theorem. In Theorem 4.3, the equivalence (3) ⇔ (4) extends (a) ⇔ (b) of[17, Theorem 1], where Y = R.
Theorem 4.3. Let X be a topological space and Y an ordered topological vector space with a positive interiorpoint. Then, the following conditions (1), (2) and (3) are equivalent:
(1) X is a cb-space.(2) For each locally upper bounded map f : X → Y , there exists a continuous map Φ(f) : X → Y such that
f � Φ(f).(3) For each upper semi-continuous map f : X → Y , there exists a continuous map Φ(f) : X → Y such
that f � Φ(f).
If we assume further that Y is a topological vector lattice, then (4) is equivalent to (1):
(4) For each locally bounded map f : X → Y , there exists a continuous map Φ(f) : X → Y such that|f | � Φ(f).
Proof. Let e be a positive interior point of Y .(1) ⇒ (2): Assume (1) and let f : X → Y be a locally upper bounded map. By (1) ⇔ (4) of Proposi-
tion 2.4, {int(f−1(ne− Y +)): n ∈ N} is an increasing open cover of X, and hence, there exists a countablelocally finite partition of unity {pn: n ∈ N} on X such that p−1
n ((0, 1]) ⊂ int(f−1(ne− Y +)) for each n ∈ N
([17, Theorem 1], [19, Theorem 1.2]). Consider the map Φ(f) : X → Y defined by Φ(f)(x) =∑∞
n=1 pn(x) ·nefor each x ∈ X. Then, Φ(f) is continuous. Also,
Φ(f)(x) =∞∑
n=1pn(x) · ne �
∞∑
n=1pn(x)f(x) = f(x)
holds for each x ∈ X, because f(x) � ne whenever pn(x) > 0. Hence, we have f � Φ(f). Thus, (2) holds.(2) ⇒ (3): The equivalence (1) ⇔ (5) of Proposition 2.4 shows that every upper semi-continuous map
f : X → Y is locally upper bounded. Thus, (2) implies (3).
K. Yamazaki / Topology and its Applications 169 (2014) 51–70 65
(3) ⇒ (1): Assume (3) and let {Un: n ∈ N} be a countable increasing open cover of X. It suffices tofind a countable refinement of {Un: n ∈ N} consisting cozero-sets (= functionally open sets [7]), see [17,19].Set n(x) = min{n ∈ N: x ∈ Un} and define a map f : X → Y by f(x) = n(x) · e for each x ∈ X.Then, f is upper semi-continuous, because f(Un(x)) ⊂ n(x) · e − Y + = f(x) − Y +. Hence, it follows from(3) that there exists a continuous map Φ(f) : X → Y such that f � Φ(f). Since Y is a Tychonoff spaceand [−e, e] is a 0-neighborhood (Proposition 2.1), take a cozero-set C of Y such that 0 ∈ C ⊂ [−e, e].Now, we have {Φ(f)−1(nC): n ∈ N} is a cozero-set cover of X. To see Φ(f)−1(nC) ⊂ Un, fix n ∈ N andlet x ∈ Φ(f)−1(nC). Then, n(x) · e = f(x) � Φ(f)(x) � ne, which implies n(x) � n since e > 0. So,x ∈ Un(x) ⊂ Un. Thus, (1) holds.
Assume, in the following, further that Y is a topological vector lattice.(3) ⇒ (4): Assume (3) and let f : X → Y be a locally bounded map. It follows from (1) ⇔ (3) of
Corollary 2.8 that there exists an upper semi-continuous φ(f) : X → Y such that |f | � φ(f). By (3), thereexists a continuous map Φ(f) : X → Y such that φ(f) � Φ(f), and we have |f | � φ(f) � Φ(f). Thus, Φ(f)is a required one in (4).
(4) ⇒ (3): Assume (4) and let f : X → Y be an upper semi-continuous map. Then, f ∨ 0 is uppersemi-continuous by (1) of Lemma 2.7. Since |f ∨ 0| = f ∨ 0, (1) ⇔ (3) of Corollary 2.8 provides that f ∨ 0is locally bounded. By (4), there exists a continuous map Φ(f) : X → Y such that |f ∨ 0| � Φ(f). Now, wehave f � f ∨ 0 = |f ∨ 0| � Φ(f). Thus, Φ(f) is a required one in (3). �5. Remarks and questions
Remark 5.1. Recall another characterization of monotonically countably paracompact (monotonically count-ably metacompact) spaces X as follows. The symbol (0,∞) stands for {r ∈ R: r > 0}.
Proposition 5.1.1. A topological space X is monotonically countably paracompact if and only if there existoperators Φ, Ψ assigning to each lower semi-continuous function f : X → (0,∞), an upper semi-continuousfunction Φ(f) : X → (0,∞) and a lower semi-continuous function Ψ(f) : X → (0,∞) with Ψ(f) � Φ(f) � f
such that Φ(f) � Φ(f ′) and Ψ(f) � Ψ(f ′) whenever f � f ′ ([8, Theorem 3], [26, Corollary 3.5]).
Proposition 5.1.2. A topological space X is monotonically countably metacompact if and only if there existsan operator Φ assigning to each lower semi-continuous function f : X → (0,∞), an upper semi-continuousfunction Φ(f) : X → (0,∞) with Φ(f) � f such that Φ(f) � Φ(f ′) whenever f � f ′ [26, Corollary 3.5].
For a function f : X → (0,∞), f is lower (resp. upper) semi-continuous if and only if the map 1/f : X → R
defined by (1/f)(x) = 1/f(x) for each x ∈ X is upper (resp. lower) semi-continuous. So, (1) ⇔ (3) ofTheorem 1.2 and (1) ⇔ (3) of Theorem 1.3 imply (in fact, and are also implied by) Proposition 5.1.1 andProposition 5.1.2, respectively. However, this technique does not work for maps f : X → Y to orderedtopological vector spaces Y . Hence, we ask:
Question 5.1.3. Let X be a topological space, and Y an ordered topological vector space with a positiveinterior point. Are the following conditions (1) and (2) equivalent?
(1) X is monotonically countably paracompact.(2) There exist operators Φ, Ψ assigning to each lower semi-continuous map f : X → Y + \ {0}, an upper
semi-continuous map Φ(f) : X → Y + \ {0} and a lower semi-continuous map Ψ(f) : X → Y + \ {0}with Ψ(f) � Φ(f) � f such that Φ(f) � Φ(f ′) and Ψ(f) � Ψ(f ′) whenever f � f ′.
Question 5.1.4. Let X be a topological space, and Y an ordered topological vector space with a positiveinterior point. Are the following conditions (1) and (2) equivalent?
66 K. Yamazaki / Topology and its Applications 169 (2014) 51–70
(1) X is monotonically countably metacompact.(2) There exists an operator Φ assigning to each lower semi-continuous map f : X → Y + \ {0}, an upper
semi-continuous map Φ(f) : X → Y + \ {0} with Φ(f) � f such that Φ(f) � Φ(f ′) whenever f � f ′.
Remark 5.2. One may ask whether or not some known set-theoretic approach like [26, Theorem 2.4] works toprovide Theorems 3.1 and 3.2. This seems to be possible only for very specific topological vector lattices Y
with positive interior points.To be convinced of this, we verify (1) ⇔ (4) of Theorem 3.1 via [26, Theorem 2.4] assuming a topo-
logical vector lattice Y with a positive interior point e further to be order complete and Dini. Here, avector lattice Y is order complete [24] if every non-empty subset A of Y , which is order bounded in Y ,has supA and inf A. A topological vector lattice Y is Dini [4,5] if every downward directed set {yα}α∈Γ
in Y , with∧
α∈Γ yα = 0, converges to 0. For a strictly increasing closed cover {Bn}n∈ω of Y , B ⊂ Y
is said to be bounded w.r.t. {Bn} [26] if B ⊂ Bn for some n ∈ ω; B(Y ; {Bn}) = {B ⊂ Y : B �= ∅,B is closed and bounded w.r.t. {Bn}} [26]; a set-valued mapping φ: X → B(Y ; {Bn}) is called locallybounded in the sense of [26] if for each x ∈ X there exist n ∈ ω and a neighborhood Ox of x such thatφ(x′) ⊂ Bn for each x′ ∈ Ox. For a set-valued mapping ϕ : X → 2Y to a vector lattice Y with
∨(ϕ(x)) for
each x ∈ X, the map∨ϕ : X → Y is defined by (
∨ϕ)(x) =
∨(ϕ(x)) for each x ∈ X.
In order to show (1) ⇒ (4) of Theorem 3.1 for this specific Y , it suffices to show that (2)′ of [26,Theorem 2.4] implies (4) of Theorem 3.1. Assume the condition (2)′ of [26, Theorem 2.4]. Let Bn = n((−e+Y +) ∩ (e − Y +)) for each n ∈ ω, then {Bn: n ∈ ω} is a strictly increasing closed cover of Y , becausee > 0. For a locally bounded map f : X → Y , the set-valued mapping φf : X → B(Y ; {Bn}) defined byφf (x) = (−|f(x)| + Y +) ∩ (|f(x)| − Y +) for each x ∈ X is locally bounded in the sense of [26]. By (2)′ of[26, Theorem 2.4], there exists a locally bounded in the sense of [26] and lower semi-continuous set-valuedmapping Φ(φf ) : X → B(Y ; {Bn}) with φf ⊂ Φ(φf ) such that Φ(φf ) ⊂ Φ(φf ′) whenever φf ⊂ φf ′ . Since Y
is order complete and each Φ(φf )(x) has an upper bound,∨
(Φ(φf )(x)) exists for each x ∈ X. Hence, by [27,Proposition 3.5] and the assumption Y being Dini, the map Ψ(f) : X → Y defined by Ψ(f) =
∨Φ(φf ) is
lower semi-continuous. Also, Ψ(f) is locally bounded with |f(x)| =∨
(φf (x)) �∨
(Φ(φf )(x)) = Ψ(f)(x) foreach x ∈ X. For a locally bounded map f ′ : X → Y with |f | � |f ′|, we have φf ⊂ φf ′ , hence Φ(φf ) ⊂ Φ(φf ′),which provides that Ψ(f) =
∨Φ(φf ) �
∨Φ(φf ′) = Ψ(f ′). Now, Ψ is a required operator in the condition
(4) of Theorem 3.1.In order to show (4) ⇒ (1) of Theorem 3.1 for this specific Y , it suffices to show that (4) of Theorem 3.1
implies (5)′ of [26, Theorem 2.4]. Let Φ be an operator as in (4) of Theorem 3.1. Put Bn = n((−e + Y +) ∩(e − Y +)) for each n ∈ ω. Then, {Bn} is a strictly increasing closed cover of Y because of e > 0. Letφ : X → B(Y ; {Bn}) be a locally bounded mapping in the sense of [26]. Since φ(x) is order boundedin Y , by the order completeness of Y , we set fφ(x) = (
∨φ(x)) ∨ (−
∧φ(x)) for each x ∈ X. Then, the
map fφ : X → Y is locally bounded. Since fφ(x) � 0 for each x ∈ X, we have |fφ| = fφ. Define aset-valued mapping Ψ(φ) : X → B(Y ; {Bn}) by Ψ(φ)(x) = (−Φ(fφ)(x) + Y +) ∩ (Φ(fφ)(x) − Y +) for eachx ∈ X. Then, Ψ(φ) is locally bounded in the sense of [26]. Since Φ(fφ) : X → Y is lower semi-continuous,Ψ(φ) is lower semi-continuous as a set-valued mapping by [5] (see also [27, Theorem 3.1]). For each locallybounded mapping φ′ : X → B(Y ; {Bn}) with φ ⊂ φ′, we have
∨φ �
∨φ′ and −
∧φ � −
∧φ′. Hence,
|fφ| = fφ = (∨
φ)∧ (−∧
φ) � (∨
φ′)∧ (−∧
φ′) = fφ′ = |fφ′ |, which implies that Φ(fφ) � Φ(fφ′), and henceΨ(φ) ⊂ Ψ(φ′). Thus, Ψ is a required operator in the condition (5)′ of [26, Theorem 2.4].
Remark 5.3. For an ordered topological vector space (Y,�), we say R is a linear positive retract of Y ifthere exists a linear positive continuous map (called a linear positive retraction) j : Y → R such that R
is a vector subspace of Y , the order � on R as a subspace of Y coincides with the usual order on R, andj(r) = r for each r ∈ R. Here, j : Y → R is positive if j(y) � 0 whenever y � 0. Simple examples of thesespaces Y are lp, c0 and C∗(X) (X �= ∅). For, identify x0 ∈ R with (x0, 0, 0, . . .) ∈ lp (or (x0, 0, 0, . . .) ∈ c0)
K. Yamazaki / Topology and its Applications 169 (2014) 51–70 67
and consider the map j : lp → R (or j : c0 → R) defined by j((xn)n∈ω) = x0. Also, identify r ∈ R withthe map 1r ∈ C∗(X), where 1r(x) = r for each x ∈ X, fix x0 ∈ X, and consider the map j : C∗(X) → R
defined by j(f) = f(x0) for each f ∈ C∗(X).
The implications (2) ⇒ (1), (3) ⇒ (1) and (4) ⇒ (1) of Theorem 3.1 (and Theorem 3.2) hold, withoutrequiring to have a positive interior point of Y , when R is a linear positive retract of Y . To show this, weprepare the following:
Fact 5.3.1. For a topological space X and a linear positive retraction j : Y → R, the following are valid.
(1) If f : X → Y is a locally upper bounded (resp. locally lower bounded) map, then j ◦ f : X → R islocally upper bounded (resp. locally lower bounded).
(2) If f : X → R is a locally upper bounded (resp. locally lower bounded) function, then f : X → R ⊂ Y
is locally upper bounded (resp. locally lower bounded) as a map with values into Y .(3) If f : X → Y is a lower semi-continuous (resp. upper semi-continuous) map, then j ◦f : X → R is lower
semi-continuous (resp. upper semi-continuous).(4) If f : X → R is a lower semi-continuous (resp. upper semi-continuous) function, then f : X → R ⊂ Y
is lower semi-continuous (resp. upper semi-continuous) as a map with values into Y .
In order to show (1) of Fact 5.3.1, let f : X → Y be a locally upper bounded map, x ∈ X andε > 0. Since j−1((−ε, ε)) is a 0-neighborhood, there exist a neighborhood Ox of x and n ∈ N such thatf(Ox) ⊂ nj−1((−ε, ε)) − Y +. Let x′ ∈ Ox, and we shall show (j ◦ f)(x′) � nε. Indeed, take y ∈ Y with−ε < j(y) < ε such that f(x′) � ny. Since j is linear positive, j ◦ f(x′) � j(ny) = n · j(y) � nε. Thus,j ◦ f : X → R is locally upper bounded.
In order to show (2) of Fact 5.3.1, let f : X → R be a locally upper bounded function, x ∈ X and U
a 0-neighborhood of Y . Take ε > 0 such that [−ε, ε] ⊂ U ∩ R, and then there exist a neighborhood Ox ofx and n ∈ N such that f(x′) � nε for each x′ ∈ Ox. Now, we have f(Ox) ⊂ nε − Y + ⊂ nU − Y +, whichprovides that f is locally upper bounded as a map with values into Y .
(3) and (4) of Fact 5.3.1 are similarly proved.Let j : Y → R be a linear positive retraction.To show (2) ⇒ (1) of Theorem 3.1 for this specific Y , without requiring to have a positive interior point
of Y , assume (2) of Theorem 3.1. Let Φ be an operator as in (2) of Theorem 3.1. By (1) ⇔ (2) of Theorem 3.1for Y = R, the following condition (5.3.2) is equivalent to the condition that X is monotonically countablyparacompact.
5.3.2. There exists an operator Ψ assigning to each locally upper bounded function g : X → R, a locallyupper bounded lower semi-continuous function Ψ(g) : X → R with g � Ψ(g) such that Ψ(g) � Ψ(g′)whenever g � g′.
Hence, it suffices to show (5.3.2). Let f : X → R be a locally upper bounded function. It follows from (2)of Fact 5.3.1 that f is locally upper bounded as a map with values into Y . Then, j ◦Φ(f) : X → R is locallyupper bounded lower semi-continuous by (1) and (3) of Fact 5.3.1. Since j is linear positive and f � Φ(f),we have f = j ◦ f � j ◦ Φ(f). If f ′ : X → R is a locally upper bounded function with f � f ′, then f ′ islocally upper bounded as a map with values into Y by (2) of Fact 5.3.1. Hence, Φ(f) � Φ(f ′), which impliesthat j ◦ Φ(f) � j ◦ Φ(f ′) since j is linear positive. Now, Ψ = j ◦ Φ is a required operator in (5.3.2). Thus,(1) of Theorem 3.1 holds.
To show (3) ⇒ (1) of Theorem 3.1 for this specific Y , without requiring to have a positive interior pointof Y , assume (3) of Theorem 3.1. Let Φ, Ψ be operators as in (3) of Theorem 3.1. Then, it suffices to show (3)
68 K. Yamazaki / Topology and its Applications 169 (2014) 51–70
of Theorem 1.2. To show this, let f : X → R be an upper semi-continuous function. Then, f : X → R ⊂ Y
is upper semi-continuous as a map with values into Y by (4) of Fact 5.3.1. Also, j ◦ Φ(f) : X → R andj ◦ Ψ(f) : X → R are lower semi-continuous and upper semi-continuous, respectively, by (4) of Fact 5.3.1.Since j is linear positive, it follows from f � Φ(f) � Ψ(f) that f = j ◦ f � j ◦ Φ(f) � j ◦ Ψ(f). Now, letf ′ : X → R be an upper semi-continuous function with f � f ′. Then, Φ(f) � Φ(f ′) and Ψ(f) � Ψ(f ′), andwhich imply j ◦Φ(f) � j ◦Φ(f ′) and j ◦Ψ(f) � j ◦Ψ(f ′) since j is linear positive. Thus, j ◦Φ and j ◦Ψ arerequired operators on (3) of Theorem 1.2.
To show (4) ⇒ (1) of Theorem 3.1 for this specific Y , without requiring to have a positive interior pointof Y , let Y be a topological vector lattice and assume (4) of Theorem 3.1. Let Φ be an operator as in(4) of Theorem 3.1. It suffices to prove that (2) of Theorem 1.2. Let f : X → R be a locally boundedfunction. Then, f : X → R ⊂ Y be locally bounded as a map with values into Y by (2) of Fact 5.3.1. SinceΦ(f) : X → Y is lower semi-continuous, j ◦ Φ(f) : X → R is lower semi-continuous by (3) of Fact 5.3.1. Wealso have j ◦Φ(f) is locally bounded by (1) of Fact 5.3.1. Since j is linear positive, for each x ∈ X, it followsfrom |f(x)| � Φ(f)(x) that |f(x)| = j(|f(x)|) � j ◦ Φ(f)(x), hence |f | � j ◦ Φ(f). For a locally boundedfunction f ′ : X → R with |f | � |f ′|, we have Φ(f) � Φ(f ′), hence which implies that j ◦ Φ(f) � j ◦ Φ(f ′)since j is linear positive. Thus, j ◦ Φ is a required operator in (2) of Theorem 1.2.
Corresponding results for Theorem 3.2 are similarly obtained.
Remark 5.4. Monotone cb-spaces are introduced in [28] by P.-F. Yan and L.-H. Xie. A space is said to bea monotone cb-space [28] if there exists an operator Φ assigning to each locally upper bounded functionf : X → R, a continuous function Φ(f) : X → R with |f | � Φ(f) such that Φ(f) � Φ(f ′) whenever|f | � |f ′|.
Related to Theorems 3.1, 3.2, 4.1, 4.2 and 4.3, we now consider the following conditions (1), (2) and (3)for a topological space X and an ordered topological vector space Y with a positive interior point. For thecondition (4), we assume further that Y is a topological vector lattice.
(1) X is a monotone cb-space.(2) There exists an operator Φ assigning to each locally upper bounded map f : X → Y , a continuous map
Φ(f) : X → Y with f � Φ(f) such that Φ(f) � Φ(f ′) whenever f � f ′.(3) There exists an operator Φ assigning to each upper semi-continuous map f : X → Y , a continuous map
Φ(f) : X → Y with f � Φ(f) such that Φ(f) � Φ(f ′) whenever f � f ′.(4) There exists an operator Φ assigning to each locally bounded map f : X → Y , a continuous map
Φ(f) : X → Y with |f | � Φ(f) such that Φ(f) � Φ(f ′) whenever |f | � |f ′|.
The implication (2) ⇒ (3) follows from the fact, which is obtained by (1) ⇔ (5) of Proposition 2.4, thatevery upper semi-continuous map is locally upper bounded.
The implication (3) ⇒ (2) also holds. To show this, let Φ be an operator as in (3). By Lemma 2.5, thereexists an operator φ assigning to each locally upper bounded map f : X → Y , an upper semi-continuousmap φ(f) : X → Y with f � φ(f) such that φ(f) � φ(f ′) whenever f � f ′. Let f : X → Y be a locallyupper bounded map. Define a map Ψ(f) : X → Y by Ψ(f)(x) = Φ(φ(f))(x) for each x ∈ X. Then, Ψ(f) iscontinuous and f � φ(f) � Φ(φ(f)) = Ψ(f). If f ′ : X → Y is a locally upper bounded map with f � f ′,then φ(f) � φ(f ′), and which provides that Ψ(f) = Φ(φ(f)) � Φ(φ(f ′)) = Ψ(f ′). Thus, Ψ is a requiredoperator in (2).
Moreover, the equivalence (2) ⇔ (4) for a topological vector lattice Y is similarly proved like (2) ⇔ (4)of Theorem 3.1. Now, we ask:
Question 5.4.1. Let X be a topological space and Y an ordered topological vector space with a positiveinterior point. Are the conditions (1) and (2) above equivalent?
K. Yamazaki / Topology and its Applications 169 (2014) 51–70 69
We also note that the implications (2) ⇒ (1), (3) ⇒ (1) and (4) ⇒ (1) of the above hold, withoutrequiring to have a positive interior point of Y , when R is a linear positive retract of Y (see Remark 5.3 fordefinition); the proof of this fact is similar to that of Remark 5.3.
Remark 5.5. The assumption ‘with a positive interior point’ of Y cannot be removed in Theorems 4.1, 4.2and 4.3. To see this, let us recall the following propositions given in [11] for κ � ω.
Proposition 5.5.1. A topological space X is κ-expandable if and only if for each upper semi-continuous mapf : X → c0(κ), there exist a lower semi-continuous map Φ(f) : X → c0(κ) and an upper semi-continuousmap Ψ(f) : X → c0(κ) such that f � Φ(f) � Ψ(f) [11, Theorem 5.1].
Proposition 5.5.2. A topological space X is almost κ-expandable if and only if for each upper semi-continuousmap f : X → c0(κ), there exists a lower semi-continuous map Φ(f) : X → c0(κ) such that f � Φ(f) [11,Theorem 5.2].
Proposition 5.5.3. A normal space X is κ-collectionwise normal and countably paracompact if and only iffor each upper semi-continuous map f : X → c0(κ), there exists a continuous map Φ(f) : X → c0(κ) suchthat f � Φ(f) [11, Corollary 5.6].
Proposition 5.5.4. A normal space X is κ-paracompact if and only if for each upper semi-continuous mapf : X → C0(κ), there exists a continuous map Φ(f) : X → C0(κ) such that f � Φ(f) ([11, Theorem 5.7],see also [21]).
Here, a space X is said to be κ-expandable (resp. almost κ-expandable) if every locally finite collection Fof closed subsets of X with |F| � κ has a locally finite (resp. point-finite) open expansion [14,25]. A space X
is a κ-collectionwise normal if every discrete collection F of closed subsets of X with |F| � κ has a disjointopen expansion (see [7]). Now, recall that c0(κ) and C0(κ) have no positive interior points (Section 2), andconsider a normal countably paracompact space X which is not κ-collectionwise normal (which is also notalmost κ-expandable), for example, the standard Bing’s example [3, Example G] for κ > ω ([7, 5.1.23], seealso [11, p. 506]). Then, Propositions 5.5.1 and 5.5.2 show that the assumption ‘with a positive interiorpoint’ of Y cannot be removed in Theorems 4.1 and 4.2, respectively. Also, either Proposition 5.5.3 orProposition 5.5.4 shows the similar fact for Theorem 4.3.
Remark 5.6. Insertion theorems are sometimes obtained by selection theorems. To demonstrate this, werecall proofs of the following insertion theorems via selection theorems; Facts 5.6.1 and 5.6.2 are obtainedby continuous selection theorems and set-valued selection theorems, respectively.
Fact 5.6.1. Every point-finite open cover U of X with |U| � κ is normal if and only if for each lowersemi-continuous map f : X → c0(κ) and each upper semi-continuous map g : X → c0(κ) with g � f , thereexists a continuous map h : X → c0(κ) such that g � h � f [11].
Fact 5.6.2. A space X is normal if and only if for each lower semi-continuous function f : X → R andeach upper semi-continuous function g : X → R with g � f , there exists a lower semi-continuous functionhl : X → R and an upper semi-continuous function hu : X → R such that g � hl � hu � f .
Indeed, the “only if” part of Fact 5.6.1 follows from the selection theorem by T. Kandô [13] andS.I. Nedev [20]. For, as in [11], consider the lower semi-continuous compact-valued mapping φ : X → 2c0(κ),defined by φ(x) = (g(x) + (c0(κ))+)∩ (f(x)− (c0(κ))+) for each x ∈ X. Then, φ has a continuous selectionh : X → c0(κ) [13,20], and h is a required map. The “only if” part of Fact 5.6.2 follows from the set-valued
70 K. Yamazaki / Topology and its Applications 169 (2014) 51–70
selection theorem by M.M. Čoban [6, Theorem 11.3]. For, consider the lower semi-continuous compact-valued mapping φ : X → 2R, defined by φ(x) = {r ∈ R: g(x) � r � f(x)} for each x ∈ X. Then, φ hasan upper semi-continuous non-empty compact-valued selection ψ : X → 2R [6]. The lower semi-continuousfunction hl =
∧ψ : X → R and the upper semi-continuous function hu =
∨ψ : X → R are required ones.
Thus, insertion theorems are frequently obtained by selection theorems.T. Kubiak established in [15, Theorem 2.5] monotone insertion theorems for monotonically normal spaces.
Monotone insertion theorems for stratifiable spaces are given by E. Lane, P. Nyikos and C. Pan [16] andC. Good and I. Stares [10]. T. Kubiak asked in [15, Question 3.8]: Is there any monotone type selection theo-rem which would characterize monotonically normal spaces? Monotone insertion theorem [15, Theorem 2.5]should be a special case of such a would-be monotone type selection theorem. Characterization of monoton-ically normal spaces X by monotone type selection theorems using maps from X to Banach spaces Y isasked in [26, Chapter 4]. We further ask:
Question 5.6.3. Is it possible to characterize monotonically normal spaces X by monotone insertion theorems(or monotone type selection theorems) using maps from X to ordered topological vector spaces Y withpositive interior points?
References
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