1 Metric spaces

1.1 Basic definitions and properties in metric spaces

Definition 1.1 A metric space is a non-empty set X together with a distance function (metric)

d : X ×X → [0,∞)

which satisfies the following conditions:

(i) d(x, y) = d(y, x)

(ii) d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

(iii) d(x, y) = 0 if and only if x = y.

Strictly speaking, the pair (X, d) is the metric space. But sometimes we just call X the metric spacewith the metric d.

Examples 1.2

(1) X = Rn with the p-metric 1 ≤ p <∞

dp(x, y) =( n∑i=1

|xi − yi|p)1/p

or the maximum-metric p =∞d∞(x, y) = max

i=1...n|xi − yi|.

Note: We require 1 ≤ p ≤ ∞ in order for the triangle inequality to be satisfied.For p = 2 we get the Euclidean metric.

Exercise: (a) Verify that limp→∞

dp(x, y) = d∞(x, y).

(b) Verify the triangle inequality for p = 1 and p =∞.

(2) For any set X 6= ∅, the discrete metric is defined by

d(x, y) =

1 if x 6= y0 if x = y

(3) Any (non-empty) subset X1 of a metric space X is a metric space again.

In particular, any X1 ⊆ Rn (X1 6= ∅) is a metric space with the Euclidean metric.

This observation gives rise to a whole multitude of examples of metric spaces.

(4) Let X = C[0, 1] be the set of all continuous functions

f : [0, 1]→ R.

This is a metric space with

d(f, g) = max |f(x)− g(x)| : x ∈ [0, 1] .

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(5) Let S be the set of all (real-valued) sequences x = xn∞n=1 (xn ∈ R). A metric is given by

d(x, y) =

∞∑i=1

1

2i· |xi − yi|

1 + |xi − yi|.

Definition 1.3 Given a metric space X with metric d, one defines the ε-neighborhood of a point x ∈ X,

Bε(x) =y ∈ X : d(x, y) < ε

.

Exercise: (a) Visualize the ε-neighborhoods Bε(x) in the case of X = R2 or X = R3 with p-metric(p = 1, 2,∞). What happens in the case of the discrete metric ?

(b) For various cases of metric spaces X ⊂ Rn (with Euclidean metric) describe the ε-neighborhoods; e.g.X= circle in R2; or sphere in R3; or line segment [0, 1] in R.

Given ε-neigborhoods, the notions of open and closed sets, interior and limit points, and closure can bedefined:

Definition 1.4 Let A ⊆ X.

(i) x is an interior point of A (written as x ∈ Ao) iffthere exists ε > 0 such that Bε(x) ⊆ A.

(ii) x is a limit point of A (or accumulation point of A; written as x ∈ A′) ifffor every ε > 0 the set Bε(x) ∩ (A \ x) is non-empty.

(iii) the closure of A is A = A ∪A′.

(iv) A is open iff every x ∈ A is an interior point, i.e., A ⊆ Ao.

(v) A is closed iff every limit point is in A, i.e., A′ ⊆ A.

Exercise: For various sets A ⊂ Rn (Rn with the Euclidean metric) decide whether the sets A are openor closed and determine Ao, A′ and A.More generally, consider various sets X ⊂ Rn as metric spaces and subsets A ⊂ X. Do the same in thissetting.

A couple of basic statements hold for these notions. As it turns out they also hold in the case of (moregeneral) topological spaces, so we will defer them to a later section. However, note that

(a): A is open iff A = Ao;

(b): A is closed iff A = A.

Furthermore, the following important statements hold in every metric space X (provide proofs).

Proposition 1.5

(a) The ε-neigborhoods Bε(x) = y ∈ X : d(x, y) < ε are always open,

(b) the set Bε(x) = y ∈ X : d(x, y) ≤ ε are always closed.

Theorem 1.6 A is open if and only if X \A is closed.

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Theorem 1.7

(a) If A and B are open, then A ∩B is open.

(b) If Aω (for ω ∈ Ω) are open sets, then⋃ω∈ΩAω is open.

Theorem 1.8

(a) If A and B are closed, then A ∪B is closed.

(b) If Aω (for ω ∈ Ω) are closed sets, then⋂ω∈ΩAω is closed.

1.2 Sequences in metric spaces and convergence

In metric spaces we can define the notion of convergence of sequences.

Definition 1.9 In a metric space X, a sequence xn∞n=1 (of elements xn ∈ X) is called convergent if thereexists x ∈ X such that

limn→∞

d(xn, x) = 0.

In this case, we write xn → x and say that xn converges to the limit x.

Note the following simple facts:

(i) the limit x of convergent sequence is unique (i.e., if xn → x and xn → y, then x = y.)

(ii) in “ε-language”, the definition takes the following form: xn → x ifffor every ε > 0 there exists an N ∈ N such that d(xn, x) < ε whenever n ≥ N .

(iii) Equivalently, xn → x iff for every ε > 0 there exists an N ∈ N such that xn ∈ Bε(x) whenever n ≥ N .

Exercises:

(i) Let X = Rd with a p-metric and let xn = (x(n)i )di=1, x = (xi)

di=1 be elements of Rd. Show that xn → x

(in p-metric) if and only if for every i = 1, . . . , d we have x(n)i → xi.

(In other words, convergence in Rd means component-wise convergence.)

(ii) Let fn, f ∈ C[0, 1] and let X = C[0, 1] have the maximum-metric. Show that fn → f (in the metricspace X) if and only if fn converges uniformly on [0, 1] to f .

(iii) What does convergence in S mean ? (This is somewhat difficult.)

(iv) What does convergence in a space with discrete metric mean ?

In metric spaces, the notions of closedness, limit points, and closure can be expressed by equivalentconditions which involve sequences. These can sometimes be quite useful.

Proposition 1.10 Let X be a metric space, and A ⊆ X.

(i) A is closed iff for every sequence xn with the property that xn ∈ A and xn → x ∈ X it follows thatx ∈ A.

(ii) x ∈ A′ iff there exists a sequence xn with the property that xn ∈ A, xn 6= x, and xn → x.

(ii) x ∈ A iff there exists a sequence xn with the property that xn ∈ A and xn → x.

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1.3 Cauchy sequences and completeness in metric spaces *

Definition 1.11 Let X be a metric space.

(i) a sequence xn∞n=1 is called a Cauchy sequence if for every ε > 0 there exists N such that

d(xn, xm) < ε for all n,m ≥ N.

(ii) The metric space is called complete if every Cauchy sequence is convergent.

Note that it is very easy to show that every convergent sequence is a Cauchy sequence, i.e., if xn → x,then xn is Cauchy. Completeness of a metric space means that the converse implication (always) holds.

Examples 1.12

(1) Rn is complete, the interval [0,∞) is complete.

(2) Q or (0, 1] are not complete (in Euclidean metric).

(3) Every space with discrete metric is complete.

(4) Let X1 ⊆ X and assume that X is a complete metric space. Then the metric space X1 is complete ifand only if X1 is a closed subset of X.

(5) The metric spaces C[0, 1] and S are complete.

1.4 Equivalent metrics

This section is meant to provide some motivation for the definition of topological spaces done in the nextchapter.

One can have different metrics on the same space X. For instance, we can consider the space Rn withany of the p-metrics or with the discrete metric. (To be precise, we obtain different metric spaces with thesame underlying set X.)

The question arises what happens with the notions of openess, closedness, interior points, limit points,and convergence. Do they depend on the underlying metric or not? The short answer is that in generalthey do depend on the metric, but sometimes they do not. A more detailed answer will involve the notionof “equivalent metrics”.

We have already seen (see Exercise after Definition 1.3) that different metrics can lead to different ε-neighborhoods. Hence given two metrics d1 and d2 we should denote the ε-neighborhoods differently,

B(i)ε (x, y) =

y ∈ X : di(x, y) < ε

i = 1, 2.

Definition 1.13 We say that two metrics d1 and d2 on X are equivalent if the following hold:

(i) For every x ∈ X and every ε > 0 there exists some δ > 0 such that B(1)δ (x) ⊆ B(2)

ε (x)

(ii) For every x ∈ X and every ε > 0 there exists some δ > 0 such that B(2)δ (x) ⊆ B(1)

ε (x)

Examples 1.14

(1) If we take X = R and d1 = discrete metric, and d2 = Euclidean metric, then these two metrics are notequivalent. Why ?

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(2) If we take X = Rn and d1 as the p1-metric and d2 as the p2 metric (where 1 ≤ p1, p2 ≤ ∞), then thesetwo metrics are equivalent. Why ?

(3) Let X = (x, y) : x2 + y2 = 1 be the unit circle in R2. We can define one metric d1 as the Euclideanmetric (inherited from R2) and another metric d2 as the “angular distance” between two points. Bothmetrics are equivalent.One can generalize this example to the sphere in R3 or even higher dimensions.

The notions of metrics being equivalent is important because of the following results:

Proposition 1.15 Let d1 and d2 be two metrics on a set X. Then the following statements are equivalent:

(a) d1 and d2 are equivalent;

(b) For every set A ⊆ X: A is open in d1-metric iff A is open in d2-metric;

(c) For every set A ⊆ X: A is closed in d1-metric iff A is closed in d2-metric;

Likewise, we have the following results.

Proposition 1.16 Let d1 and d2 be two equivalent metrics on a set X. Then

(a) x is a limit point of a subset A in the d1-metric iff it is a limit point in the d2-metric;

(b) x is an interior point of a subset A in the d1-metric iff it is a interior point the d2-metric;

(c) xn converges to x in the d1-metric iff xn converges to x in the d2-metric.

What we see from the above is that for equivalent metrics on a set X, the notions of being an open setor a closed set, as well as the notions of limit points, interior points, and convergence amount to the same.Such notions or properties (which remain the same for equivalent metrics) are called topological notions orproperties.

Example 1.17 Not all notions for metric spaces are topological notions. For instance, the notion ofcompleteness and of being a Cauchy sequence are not. The notion of a bounded set is also not a topologicalnotion.Consider, e.g., X = (0, 1] = x ∈ R : 0 < x ≤ 1 with the equivalent metrics

d1(x, y) = |x− y|, d2(x, y) =

∣∣∣∣ 1x − 1

y

∣∣∣∣ .One can show:The sequence xn = 1

n is Cauchy in the d1-metric, but not in the d2-metric. In fact, the space X with thed1-metric is not complete, while the space X with the d2-metric is complete.

The notion of topological spaces will generalize the notion of a metric space. We will give up on the“quantitive characterization” that the metric provides. However, in topological spaces we will retain the“qualitative characterizations” that are inherent in the topological properties.

Every metric space is a topological space. In fact, spaces with equivalent metric will give rise to the sametopological space. On the other hand, there are topological spaces that cannot arise from metric spaces.

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2 Topological spaces

2.1 Motivation and definitions: topological spaces - bases and topologies

Topological spaces are a generalization of metric spaces. This means, every metric space is also a topologicalspace but not necessarily vice versa.

In a topological space we will abandon the notion of a metric and of an ε-neighborhood. However, wewill keep the notion of open sets. In fact, the definition of a topological space will be based on it. We willalso retain the notions of closed sets, limits points, interior points and the closure of a set.

Recall that in a metric space X, the empty set ∅ and X are always open. Also recall Theorem 1.7regarding unions and intersections of open sets. We now give the definition of a topological space. ThereinP (X) stands for the power set (= set of all subsets) of X.

Definition 2.1 A topological space is a set X 6= ∅ together with subset τ ⊆ P (X) satisfying

(i) ∅, X ∈ τ ;

(ii) if A,B ∈ τ , then A ∩B ∈ τ ;

(iii) if Aω (for ω ∈ Ω), then⋃ω∈Ω

Aω ∈ τ .

The set τ is called the topology on X. The sets A ∈ τ are by definition called the open sets in this topologicalspace.

In other words, in a topological space we prescribe the set of open sets (the topology τ), and only requirethat the conditions (i)–(iii) are fulfilled.

Every metric space is a topological space if we consider as τ the set of all open sets (as defined for metricspaces in Definition 1.4). Equivalent metrics on a set X give rise to the same topology on X.

We will give examples of topological spaces below. In some instances it is possible to define τ direct. Inother instances it is more difficult and requires the notion of a base of a topological space. The notion of abase generalizes the notion of the set of all ε-neighborhoods in a metric space.

In order to motivate the definition a base of a topological space, we go back to metric spaces let us recallsome properties of the set of ε-neighborhoods in metric spaces.

Proposition 2.2 Let X be a metric space.

(a) A set A is open iff A can be written as the union of certain ε-neighborhoods, i.e.,

A =⋃ω∈Ω

Bεω (xω)

(b) If z ∈ Bε1(x1) ∩Bε2(x2), then there exists δ > 0 such that

z ∈ Bδ(z) ⊆ Bε1(x1) ∩Bε2(x2)

(c) Every x ∈ X is contained in some ε-neighborhood.

Proof. (c) is trivial and (b) is easy to prove. As for (a), we have to check both directions: If A is writtenin this way, then A is open. (Why ?) Conversely, assume A to be open, then for every x ∈ A we find εx > 0such that Bεx(x) ⊆ A. Now verify that

A =⋃x∈A

Bεx(x).

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6

Part (b) and (c) of the above proposition above will be taken as a motivation for the definition of a baseof a topological space.

Definition 2.3 Let X 6= ∅. A set b ⊆ P (X) is called a base of a topology on X if the following is satisfied.

(i) If B1, B2 ∈ b and z ∈ B1 ∩B2, then there exists B ∈ b such that

z ∈ B ⊆ B1 ∩B2.

(ii) For every x ∈ X there exists B ∈ b such that x ∈ B.

Examples 2.4

(1) Let X be a metric space. Then a base is given by

b1 =Bε(x) : x ∈ X, ε > 0

.

There are other bases, e.g.,

b2 =B1/n(x) : x ∈ X, n ∈ N

.

(2) Let X = Rd. Besides what was stated in (1), another possible base is

b =B1/n(x) : x ∈ Qd, n ∈ N

.

Now motivated by part (a) of Proposition 2.2 we can now establish that we can use the notion of a basein order to define a topology (and thus a topological space).

Theorem 2.5 Let X 6= ∅, and let b be a base for a topology on X.

τ =A =

⋃ω∈Ω

Bω : index set Ω, Bω ∈ b for all ω ∈ Ω.

Then is a topology on X.

In this case, we also say that b is a base for the topology τ or for the topological space (X, τ).

Proof. We have to verify (i)–(iii) in Definition 2.1 of a topological space. Note that ∅ ∈ τ since, formally,we obtain it as the ‘empty union’ over the index set Ω = ∅. Furthermore X ∈ τ because of property (ii) inDefinition 2.3. Property (iii) is trivially fulfilled, while (ii) requires some more work (which we leave as anexercise). 2

The following clarifies further the relationship between a topology and the base for this topology.

Proposition 2.6 Let (X, τ) be a topological space and let b be a base for the topology τ . Then b ⊆ τ and

• for every A ∈ τ and every x ∈ A there exists U ∈ b such that x ∈ U ∈ A.

In many textbooks, the previous property is taken as the definition for “b to be the base of the topologyτ”. Notice that here, the topology τ is given, whereas in Definition 2.3 it is absent and thus b can be usedto define a topology (as is done Theorem 2.5).

Example 2.7 Consider a metric space X with the bases (see above)

b1 =Bε(x) : x ∈ X, ε > 0

, b2 =

B1/n(x) : x ∈ X, n ∈ N

.

Both bases give the same topology, the so-called metric topology.

How one can actually see whether two bases give the same topology ? There is some criterion, which wewill discuss later.

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2.2 Some examples of topological spaces

Examples 2.8

(1) Every metric space is a topological space with the “metric-topology”.

(2) For any set X 6= ∅, the discrete topology is defined by τ = P (X). It also arises from a metric spacewith discrete metric.

(3) For any set X 6= ∅, the trival topology is τ = ∅, X.

(4) Let X = 1, 2. Then τ = ∅, 1, 1, 2 is a topology.

(5) Let X = 1, 2, 3. Then τ = ∅, 1, 1, 2, 1, 3, 1, 2, 3 is a topology.

(6) Let X = R and τ = ∅, X ∪ (−∞, a) : a ∈ R .

(7) Let X be an infinite set. The finite complement topology is

τ = ∅ ∪ U ⊆ X : X \ U is finite

(8) Let X = R and b = (a, b] : a < b. The topology arising from this base is called the lower-limittopology.

(9) Let X = R and b = (a, b) : a < b. This gives the “usual” topology on R.

(10) Let X = R and b = (a, b] : a < b, a, b ∈ Q. This gives yet another (different) topology on R.

(11) The real line with double zero.Let X = R ∪ 0′, where 0′ is just a symbol (denoting a “copy” of the usual zero).Put Bε(x) = (x− ε, x+ ε) and Bε(0

′) = (−ε, 0) ∪ 0′ ∪ (0, ε) and consider the base

b = Bε(x) : x ∈ R ∪ 0′ .

This gives a topology on X. Note that X is not a metric spaces. (We used Bε(x) only for convenienceof notation.)

2.2.1 Subspace topology

Given a topological space X with topology τ and any subset Y ⊂ X (Y 6= ∅), this set Y naturally becomesa topological space as well. Namely, one can define the so-called subspace topology on Y as follows

τY = U ∩ Y : U ∈ τ .

One can show that if τ satisfies the axioms for a topology, then so does τY (see Definition 2.1).Hence a set V ⊆ Y is open in Y if and only if it can be represented as an intersection V = U ∩ Y where

U is an open set in X.There is also a relationship to bases of a topological space. If b is a base for a topological space X and

Y ⊂ X, thenbY = U ∩ Y : U ∈ b, U ∩ Y 6= ∅

is a base for Y with the subspace topology. (Here we also consider all intersection, but discard the setwhenever it is empty.)

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2.3 Hausdorff spaces

Any open set U in a topogical space (i.e., U ∈ τ) for which x ∈ U will be called an (open) neighborhood of x.

Definition 2.9 A topological space is called Hausdorff (or a Hausdorff space) if for all x, y ∈ X with x 6= ythere exists U, V ∈ τ such that

x ∈ U, y ∈ V, U ∩ V = ∅.

The Hausdorff property is also called a separation property. (There exist other kinds of separation prop-erties as well.) Basically it says that any two distinct points can be separated by two disjoint neighborhoods.

Proposition 2.10 Every metric space is a Hausdorff topological space.

Proof. Given x 6= y we have d(x, y) > 0. We put ε = 1/2 ·d(x, y) and now take as neighborhoods U = Bε(x)and V = Bε(y). One can show that U ∩ V = ∅. 2

Examples 2.11

(1) Any space with discrete topology is Hausdorff. Any subset of X ⊆ Rn with the ususal topology(induced by the metric) is Hausdorff. This is because these are metric spaces.

(2) The trivial topology on any set |X| ≥ 2 is not Hausdorff.

The finite complement topology on any infinite set X is not Hausdorff.

The “real line with double zero” is also not Hausdorff.

(3) The spaces defined in (8) and (10) of Example 2.8 are Hausdorff.

2.4 Further basic topological notions

Besides the notion of an open set (which the very definition of topological spaces is based upon) one has alsoother notions, which we already know from metric spaces. There are several (equivalent) ways of definingthem. We choose the following ones.

Definition 2.12 Let X be a topological space with topology τ .

(i) A subset A ⊆ X is called closed if its complement X \A is open;

(ii) x is an interior point of A (written as x ∈ Ao) if there exists U ∈ τ with x ∈ U ⊆ A;

(iii) x is a limit point of A (written as x ∈ A′) if for every U ∈ τ with x ∈ U we have

U ∩ (A \ x) 6= ∅;

(iv) the closure is defined as A = A ∪A′.

Now one can show, for instance, the following statements:

Proposition 2.13

(a) A is open iff A ⊆ Ao iff A = Ao;

(b) A is closed iff A′ ⊆ A iff A = A;

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(c) x ∈ A iff for every U ∈ τ with x ∈ U we have

U ∩A 6= ∅;

(d) Ao is always open; A′ and A are always closed;

(e) (X \A)o = X \A and (X \A) = X \Ao;

(f) Ao is the largest open set contained in A;

(g) A is the smallest closed set containing A;

2.4.1 One important “technical” point of attention *

In many statements (but not all !) of the kind as above one can replace “for all U ∈ τ” by “for all U ∈ b”and one can replace “there exists U ∈ τ” by “there exists U ∈ b”, where b is a base for the topology τ . Wewill give examples illustrating this in a few moments.

Before we mention the reason why this is possible, namely the following relationship between a base andits topology. The reason that we can make this replacement is Proposition 2.6 saying that

• for every U ∈ τ with x ∈ U there exists V ∈ b with x ∈ V ⊆ U .

whenever b is the base for the topology τ . This replacement is very useful because in many instances oftopological spaces, the base b is given explicitly, whereas τ is not.

We give an illustration of this replacement by equivalently expressing the conditions x ∈ Ao and x ∈ A:

Proposition 2.14 Let X be a topological space with topology τ and a base b, let A ⊂ X and x ∈ X.

(i) there exists U ∈ τ with x ∈ U ⊆ A iff there exists V ∈ b with x ∈ V ⊆ A.

(ii) for every U ∈ τ with x ∈ U we have U ∩A 6= ∅ ifffor every U ∈ b with x ∈ U we have U ∩A 6= ∅.

Proof. (i) the ⇐ direction is trivial since b ⊆ τ (we can take U = V ). As for the ⇒ direction, theassumption is that there exists U ∈ τ with x ∈ U ⊆ A. But by the lemma we obtain V ∈ b with x ∈ V ⊆ U .Thus x ∈ V ⊆ U ⊆ A.

(ii) the ⇒ direction is trivial since b ⊆ τ . As for the ⇐ direction let U ∈ τ with x ∈ U be given. Applingthe lemme we obtain V ∈ b with x ∈ V ⊆ U . Now apply the assumption (with U replaced by V ). (Theassumption is that for every V ∈ b with x ∈ V we have V ∩A 6= ∅.) Hence we have V ∩A 6= ∅. Since V ⊆ U ,it follows that U ∩A 6= ∅. 2

2.5 Equivalent bases for topological spaces *

Let us go back to version 1 of the definition of a topological space. Let X be a nonempty set and b1 be abase for it. This base determines a topology τ1 for X and makes X a topological space. Now assume thatwe are given another base b2, which of course determines a topology τ2. This topology may or may not bethe same as the first one. The question is how can we find out whether the two topologies are the same, i.e.,whether τ1 = τ2.

If they are the same, the two base give the same topological space. Otherwise we get two differenttopological spaces (even though the set X is the same).

Here is a criterion.

Proposition 2.15 Let X be a non-empty set and let b1 and b2 be two bases for two topologies τ1 and τ2.Then τ1 = τ2 if and only if

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(i) for every U ∈ b1 and every x ∈ U there exists V ∈ b2 such that x ∈ V ⊆ U ;

(ii) for every U ∈ b2 and every x ∈ U there exists V ∈ b1 such that x ∈ V ⊆ U .

For a proof of this proposition one can use Lemma ??.

There is a certain similarity to the notion of equivalent metrics (see Section 1.4). Indeed this can bemade rigorous and leads to the following observation concerning metric spaces and their topologies.

Proposition 2.16 Let d1 and d2 be two metrics on a set X, which give rise to two topologies τ1 and τ2.Then τ1 = τ2 if and only if the two metrics are equivalent.

In other words, two equivalent metrics give rise to the same topological space. This means that X withd1 and X with d2 may be two different metric spaces (because d1 6= d2). However, as topological spaces theywill be the same.

Example 2.17

(1) Review Examples 2.4 in view of the above propositions.

(2) X = Rd with any p-metric (1 ≤ p ≤ ∞) will give the same topology (called the usual or standardtopology on Rd).

(3) X = Rd with the discrete metric with give a different topology (the discrete topology).

(4) For X = R, the basesbusual = (a, b) : a < b

blowlim = (a, b] : a < b

blowlim,Q = (a, b] a < b, a, b ∈ Q

give three different topologies, the usual one, the lower limit, and another one. The base

b′usual = (a, b) : a < b, a, b ∈ Q ,

however, gives the usual topology on R again.

2.6 Convergence of sequences*

One way to get some clearer idea of what a topology on a set X “means”, is by looking at the concept ofconvergence.

Definition 2.18 Let X be a topological space. Then a sequence xn∞n=1 converges to a point x ∈ X if forevery U ∈ τ with x ∈ U there exists N such that xn ∈ U for all n ≥ N .

In other words, for every neighborhood U of x, all but finitely many of elements of the sequence shouldbelong to U .

Examples 2.19

(1) Suppose X has the trivial topology. Then xn → x does always hold.

(2) Suppoe X has the discrete topology. Then xn → x iff there exists N such that xn = x for all n ≥ N .

(3) Let X = 1, 2, τ = ∅, 1, X. Then xn → 2 hold always; and xn → 1 holds iff xn = 1 for n ≥ N .

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(4) Let X = R with lower limit topology. Then xn → x iff for every ε > 0 there exists N such that

x− ε < xn ≤ x, for all n ≥ N.

(The sequence convergences from below to x; that’s why the name “lower limit”.)

(5) Let X = R with τ = ∅,R ∪ (−∞, a) : a ∈ R. Then xn → x iff

lim supn→∞

xn ≤ x.

(6) Let X be infinite with finite complement topology. Then xn → x iff every a ∈ X \ x occurs at mostfinitely many times in the sequence xn.

(7) Let X be an uncountable set with the countable complement topology,

τ = ∅ ∪A ⊂ X : X \A is finite or countable

Then xn → x iff there exists N such that xn = x for all n ≥ N .

(Remark: This shows that in view of convergence of sequences, the countable complement topologyand the discrete topology appear to be the same. However, they are not the same (for X beinguncountable). This indicates some limitations in using convergence of sequences for the description ofa topology.)

Some of the examples show that the limit of a sequence need not be unique. This can only happen innon-Hausdorff spaces.

Proposition 2.20 Let X be a Hausdorff space and let xn → x and xn → y. Then x = y.

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3 Continuous functions and homeomorphisms

3.1 The notion of continuity

Let us first recall what continuity means in metric spaces. There is a global and a local version. The localversion just means continuity at a point. The global version means that the function is continuous at everypoint of the domain. We will first focus on the local version.

Definition 3.1 Let X and Y be metric spaces and f : X → Y . The function f is continuous at a point x0

if for every ε > 0 there exists δ > 0 such that for all x ∈ X,

dX(x, x0) < δ implies dY (f(x), f(x0)) < ε.

One can rephrase this implication involved here by noting that dX(x, x0) < δ iff x ∈ Bδ(x0) and thatdY (f(x), f(x0)) < ε iff f(x) ∈ Bε(f(x0)) using ε- and δ-neighborhoods in Y and X, respectively. Further,using the notion of a pre-image of a set A ⊂ Y ,

f−1(A) = x ∈ X : f(x) ∈ A

the above amounts to:

For every ε > 0 there exists δ > 0 such that

Bδ(x0) ⊆ f−1(Bε(f(x0))).

Motivated by this one gives the following definition of continuity at a point for functions in topologicalspaces:

Definition 3.2 Let X and Y be topological spaces and f : X → Y . Then f is continuous at a point x0 ∈ Xif for every U ∈ τY with f(x0) ∈ U there exists V ∈ τX with x0 ∈ V such that

V ⊆ f−1(U).

In words: the pre-image of every neighborhood U of f(x0) contains some neighborhood V of x0. It ispossible to replace in this definition the topologies τX and τY by corresponding bases bX and bY . (Theresulting statement is equivalent – give a proof of it; see also Section 2.4.1).

Note that for continuity at a point, the pre-image f−1(U) need not be open. (Find some example). Thischanges when considering functions which are continuous at every point x0 ∈ X.

Theorem 3.3 Let X and Y be topological spaces and f : X → Y . Then f is continuous at every pointx ∈ X if and only if for every U ∈ τY , the pre-image f−1(U) belongs to τX .

In words: the pre-image of every open set has to be open. (Sometimes this statement is given as thedefinition of the continuity of a function, in particular, when the issue of continuity at point is not discussed.)If a function is continuous at every point x ∈ X we also say that f is continuous (on X).

Theorem 3.4 Let X and Y be topological spaces and f : X → Y . Then f is continuous on x ∈ X if andonly if the pre-image of every closed set is closed.

Proof. Use the relation f−1(Y \A) = X \ f−1(A). 2

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3.2 Examples and basic properties

Examples 3.5

(1) Let X = Y = R and let f(x) = x. As topologies τX and τY consider the usual topology, the discretetopology and the lower limit topology (in any combination). Examine whether f is continuous.

(2) Let X = R2 and Y = R with standard topologies. Let f(x, y) = x. Show that f is continuous.

(3) Let X = R and Y = R2 with standard topologies. Let f(x) = (x, 0). Show that f is continuous.

(3) Let X = (0,∞) and Y = R2 with standard topologies. Let f(x) = (x, sin(1/x)). Show that f iscontinuous.

(4) There exist continuous functions f : [0, 1] → [0, 1]2 (usual topologies) which are surjective, so-calledspace-filling curves, e.g., Peano’s space filling curve.

Proposition 3.6 The composition of two continuous functions is continuous.

Lemma 3.7 (Gluing lemma) Let X be a topological space such that X = A ∪ B with A and B beingclosed. Let f : A → Y and g : B → Y be continuous functions such that f(x) = g(x) for x ∈ A ∩ B. Thenthe function h : X → Y ,

h(x) =

f(x) x ∈ Ag(x) x ∈ B

is continuous on X.

3.3 Homeomorphisms

Definition 3.8 Let X and Y be topological spaces. A function f : X → Y is called a homeomorphism(between X and Y ) if f is continuous and bijective and its inverse function f−1 is also continuous.

If there exists a homeomorphism between X and Y , then X and Y are called homeomorphic.

The relation that X is homoemorphic to Y is an equivalence relation. (Why ?)The notion of homeomorphism and homeomorphy is at the very heart of topology. The basic question

one wants to answer is: Are two given spaces homeomorphic to each other ?If two spaces are homeomorphic to each other one usually constructs a function and proves that it is a

homeomorphism.However, if two spaces are not homoemorphic to each other, then the strategy is to find a topological

invariant and to show that this invariant is different for the spaces under consideration. (We will discussthis in later sections.)

Examples 3.9

(1) The spaces X = Y = 1, 2 with topologies τX = ∅, 1, 1, 2 and τY = ∅, 2, 1, 2 are differenttopological spaces (X = Y , but τX 6= τY ). However, they are homeomorphic to each other.

(2) The spaces (0, 1), (0,∞) and R (with standard topologies) are homeomorphic to each other.

(3) X = [0, 1]2 and Y = (x, y) : x2 + y2 ≤ 1 are homeomorphic to each other.

(4) X = (0,∞) and Y = (x, sin(1/x)) : x > 0 ⊆ R2 are homeomorphic to each other.

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(5) X = (0, 1] and Y = [1,∞) are homeomorphic to each other.

Remark: As metric spaces (with Euclidean metric), however, one is complete the other is not. Thisindicates that the notions of Cauchy sequences and completeness cannot be reasonably defined fortopological spaces in general.

(6) The sphere S2 = (x, y, z) : x2 + y2 + z2 = 1 and the so-called one-point compactification of R2,

R2 := R2 ∪ ∞

are homeomorphic. Here R2 has the topology given by the base

b = Bε(x) : x ∈ R2, ε > 0 ∪ BR(∞) : R > 0

where BR(∞) := ∞ ∪ x ∈ R2 : ‖x‖ > R.

3.4 Quotient topology

Let X be a nonempty set and ∼ be an equivalence relation on it. The equivalence classes are defined by

[x]∼ = y ∈ X : x ∼ y .

The equivalence classes partition X into disjoint subsets. Every x ∈ X belongs to some equivalence class(namely x ∈ [x]∼). Any two equivalence classes [x]∼ and [y]∼ are either the same ([x]∼ = [y]∼) or disjoint([x]∼ ∩ [y]∼ = ∅). We denote by X/ ∼ the set of all equivalence classes.

Examples 3.10

(1) Let X = Z and x ∼ y iff 3|(x− y). Then we have three equivalence classes

(X/ ∼) =

[0]∼, [1]∼, [2]∼

.

(2) Let X = R and x ∼ y iff x− y ∈ Z. Then any equivalence class looks like

[x]∼ = x+ n : n ∈ Z.

We can “parameterize” the set of all equivalence classes by a parameter t ∈ [0, 1). By this we meanthat there is a bijective function

φ : [0, 1)→ X/ ∼

which is given by φ(t) = [t]∼. Hence, as sets, X/ ∼ and [0, 1) are “the same”. This is of course onlythe simplest parameterization. There are infinitely many different ones.

Now assume that X is not only a set, but a topological space. The question arises whether X/ ∼ is atopological spaces as well. The answer is yes, and it leads us to the definition of the quotient space X/ ∼with the quotient topology.

Definition 3.11 Let X be a topological space with topology τ and let ∼ be an equivalence relation. Considerthe (so-called) quotient map

p : x ∈ X 7→ [x]∼ ∈ X/ ∼ .

Then we define the quotient topology τ∼ ⊆ P ((X/ ∼)) as follows: for U ⊆ X,

U ∈ τ∼ iff p−1(U) ∈ τ.

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It is straightforward to see that τ∼ in indeed a topology. Moreover, the quotient map is continuous fromX to X/ ∼. In fact, τ∼ is the largest topology (among all topologies on X/τ) such that p : X → X/ ∼ iscontinuous.

In practice it may be quite difficult to describe the quotient topology in explicit terms. Basically one hasto take the following into account.

Definition 3.12 A subset A ⊂ X (where X has an equivalence relation ∼) is called saturated if x ∈ A andx ∼ y implies y ∈ A.

This means, with a single element x belonging to A, all the elements of its equivalence class [x]∼ mustbelong to A as well.

This definition is relevant because the quotient map p : X → X/ ∼ has the property that for every setU ⊂ X/ ∼, the pre-image p−1(U) is saturated. Coming back to the definition of the quotient topology weobserve that U ⊆ X/ ∼ is open iff A = p−1(U) is open in X and saturated. Thus in order to describe allsets U which are open in the quotient space X/ ∼ one needs to find all open and saturated sets A ⊆ X. Infact, one can show that there is a one-to-one correspondence between open sets U in X/ ∼ and open andsaturated sets A ⊂ X.

Example 3.13 Let X = R2 with the standard topology and let (x1, y1) ∼ (x2, y2) iff x1 = x2. Theequivalence classes are the vertical lines in R2. Every vertical line is uniquely described by its x-coordinate(or the point of intersection of the vertical line with the x-axis.) Thus as sets R2/ ∼ and R can be identified.The bijection φ : R→ R2/ ∼ is

φ : x 7→ [(x, 0)]∼ = (x, y) : y ∈ R .

A set A in R2 is saturated if it is the union of certain vertical lines. One can show that A ⊂ R2 is saturatedand open iff there exists an open set U ⊂ R such that A = (x, y) : x ∈ U, y ∈ R = U × R. (This issomewhat technical.)

We have the quotient topology on R2/ ∼, but we want to know what is the topology on R such that φ isa homeomorphism. (Answering identified R2/ ∼ with R not only as a set, but also as a topological space.)So the question is, for which sets U ∈ R is the set (p−1 φ−1)(U) open and saturated. A computation yields(p−1 φ−1)(U) = U × R. As noted above this set is open (in R2) iff U is open in R. Thus R with thestandard topology makes φ a homeomorphism.

To summarize we say that R2/ ∼ is homoemorphic to R (with standard topology) via the homeomorphismφ.

Let us present more examples, but omit all the details.

Examples 3.14

(1) Let X = R with equivalence relation x ∼ y iff x − y ∈ Z. Then X/ ∼ is homoeomorphic to a circleS1 = (cos(2πφ), sin(2πφ)) : φ ∈ R . Here S1 has the subspace topology of R2.

The homeomorphism is given by φ : S1 → X/ ∼ by

φ : (cos(2πφ), sin(2πφ)) 7→ [φ]∼ = φ+ n : n ∈ Z .

(2) Let X = R2 with equivalence relation (x1, y1) ∼ (x2, y2) iff x1 − x2 ∈ Z and y1 = y2. Then X/ ∼ ishomeomorphic to the (infinite) cylinder S1 × R.

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(3) Let X = R2 with equivalence relation (x1, y1) ∼ (x2, y2) iff x1 − x2 ∈ Z and y1 − y2 ∈ Z. Then X/ ∼is homeomorphic to a torus S1 × S1.

(4) Let X = R2 \ 0 with equivalence relation (x1, y1) ∼ (x2, y2) iff (x1, y1) = 2k(x2, y2) for some k ∈ Z.Then X/ ∼ is homeomorphic to a torus S1 × S1.

(5) Let X = R with standard topology and an equivalent relation given by the equivalence classes,

[−] = x ∈ R : x < 0, [0] = 0, [+] = x ∈ R : x > 0.

Hence X/ ∼= [−], [+], [0]. Then the quotient topology is the following non-Hausdorff topology

τ∼ =∅, [+], [−], [+], [−], [+], [−], [0]

.

(6) Let X = [0, 1] with an equivalence relation that identifies 0 and 1. This means:

x ∼ y iff (x = y) or (x = 1 and y = 0) or (x = 0 and y = 1).

Then X/ ∼ is homeomorphic to the circle S1.

(7) Let X = D = (x, y) : x2 + y2 ≤ 1 with an equivalence relation that identifies all points on the unitcircle. This means:

(x1, y1) ∼ (x2, y2) iff (x1, y1) = (x2, y2) or x21 + y2

1 = x22 + y2

2 = 1.

Then X/ ∼ is homeomorphic to the sphere S2.

(8) Let X = [0, 1]×[0, 1] with an equivalence relation that identifies points (0, y) and (1, y) for all 0 ≤ y ≤ 1.This means

(x1, y1) ∼ (x2, y2) iff (x1, y1) = (x2, y2) or (y1 = y2 and x1, x2 ∈ 0, 1).

The equivalence classes are the following ones:

– [(x, y)] = (x, y) for 0 < x < 1 and 0 ≤ y ≤ 1;

– [(0, y)] = (0, y), (1, y) for 0 ≤ y ≤ 1.

Then X/ ∼ is homeomorphic to a finite cylinder S1 × [0, 1].

(9) Let X = [0, 1]×[0, 1] with an equivalence relation that identifies points (0, y) and (1, y) for all 0 ≤ y ≤ 1,and which identifies all points (x, 0) and (x, 1) for all 0 ≤ x ≤ 1 This means

(x1, y1) ∼ (x2, y2) iff (x1, y1) = (x2, y2) or (y1 = y2 and x1, x2 ∈ 0, 1) or

(x1 = x2 and y1, y2 ∈ 0, 1) or (x1, x2, y1, y2 ∈ 0, 1)

– [(x, y)] = (x, y) for 0 < x < 1 and 0 < y < 1;

– [(0, y)] = (0, y), (1, y) for 0 < y < 1;

– [(x, 0)] = (x, 0), (x, 1) for 0 < x < 1;

– [(0, 0)] = (0, 0), (0, 1), (1, 0), (1, 1).

Then X/ ∼ is homeomorphic to a torus S1 × S1.

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4 Topological properties

A topological property (of a topological space) is a property which remains invariant under homeomorphism.More precisely:

If X and Y are topological spaces which are homeomorphic to each other and if X has a property P ,then Y has also the property P .

For instance, the property of being a Hausdorff topological space is a topological property.(Try to give a proof.)

The notion of a topological property is useful because of the following:If X has a topological property P and Y has not the topological property P , then X and Y are nothomeomorphic to each other.Hence topological properties help us to distinguish topological spaces up to homeomorphism.

In what follows we consider several examples of other topological properties.

4.1 Fixed-point property

Definition 4.1 A topological space X has the fixed-point property if for every continuous function f : X →X, there exists x0 ∈ X such that f(x0) = x0.

Theorem 4.2 The fixed-point property is a topological property.

Proposition 4.3 The interval [0, 1] has the fixed-point property.

The proof is based on the intermediate value theorem.

The following is known as Browder’s fixed point theorem and its proof is much more complicated.

Theorem 4.4 For d ≥ 1, the space closed ball X = x ∈ Rd : ‖x‖ ≤ 1 has the fixed-point property.

Examples 4.5 The following spaces do not have the fixed-point property:

(0, 1), [0, 1), S1(circle), Rd, S2(sphere), S1 × S1 = T(torus), D(open disk)

To see this, give examples of continuous functions on these spaces without a fixed point.

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4.2 Path-connectedness

Definition 4.6 A topological space X is called path-connected if for every x0, x1 ∈ X there exists a contin-uous function f : [0, 1]→ X such that f(0) = x0 and f(1) = x1.(The image of f is called the path connecting x0 and x1.)

A subset A of a topological space X is path-connected if A as a topological space with the subspacetopology is path-connected. This means that A ⊆ X is connected if for x0, x1 ∈ A there exists a continuousfunction f : [0, 1]→ A such that f(0) = x0 and f(1) = x1.

Examples 4.7

(1) The subsets of R (with standard topology) are path-connected:

[0, 1], (0, 1), [0, 1)

(2) The subsets of R2 (with standard topology) are path-connected:

[0, 1)× [0, 1), x ∈ R2 : 1 < ‖x‖ < 2, (x, sin(1/x) : x > 0

(3) The following subsets of R are not path-connected

[0, 1) ∪ (1, 2], 0, 1

(4) The following subset of R is not path-connected

(x1, x2) : |x1| > 1

(5) The topological space X with discrete topology (and |X| ≥ 2) is not path-connected.

Proposition 4.8 Let X and Y be topological spaces and g : X → Y be continuous. If X is path-connected,then the image A = f(X) is a path-connected subset of Y .

Corollary 4.9 Path-connectedness is a topological property.

Example 4.10 The spaces (0, 1), [0, 1), [0, 1] and S2 are not homeomorphic to each other. Though all ofthen are path-connected, one can look at the following (topological) properties:(a) There exists one point x1 such that X \ x1 is path-connected.(b) There exists two points x1 6= x2 such that X \ x1 and X \ x2 are path-connected.

4.3 Connectedness

There exists a related but different notion called connectedness. It is best defined by its negation, discon-nectedness.

Definition 4.11 A topological space X is called disconnected if there exists open, non-empty and disjointsets A and B such that X = A ∪B.A topological space X is called connected if it is not disconnected.

We can say that X is connected iff whenever X = A ∪ B with A,B being open and A ∩ B = ∅ it mustfollows that A = ∅ or B = ∅.

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Proposition 4.12 A topological space X is connected iff X and the empty set are the only subsets of Xwhich are both open and closed.

A subset C of a topological space X is called disconnected if C as a topological space with subspacetopology is disconnected.

Proposition 4.13 A subset C of a topological space X is disconnected iff there exists open subset U and Vof X such that U ∩ C 6= ∅, V ∩ C 6= ∅, C = (U ∪ V ) ∩ C, and U ∩ V ∩ C = ∅.

Proposition 4.14 Let X and Y be topological spaces and g : X → Y be continuous. If X is connected, thenthe image A = f(X) is a connected subset of Y .

Corollary 4.15 Connectedness is a topological property.

There is a relationship between connectedness and path-connectedness.

Proposition 4.16 If X is path-connected, then X is connected.

The converse is not true.

Example 4.17 Consider the topologist’s sine curve

S = (x, sin(1/x) : x > 0

which is path-connected, hence connected. Its closure (in R2)

S = S ∪ (0, y) : −1 ≤ y ≤ 1

is also connected. This follows from the lemma below. On the other hand, S is not path-connected. Theproof that S is not path-connected is somewhat difficult and technical.

Lemma 4.18 Let A and B be subsets of a topological space X such that A ⊆ B ⊆ A. If A is connected,then B is also connected.

Proof. Assume B is disconnected. Then there exists open U, V such that

U ∩B 6= ∅, V ∩B 6= ∅, B = (U ∪ V ) ∩B, U ∩ V ∩B = ∅.

As A ⊆ B we obtain A = (U ∪V )∩A, U ∩V ∩A = ∅. But then U ∩A = ∅ or V ∩A = ∅ because otherwiseA would be disconnected.

Assume wlog. V ∩A = ∅. Equivalently, A ⊆ X \ V . Taking the closure and using that V is open we get

A ⊆ X \ V = X \ V,

which means A ∩ V = ∅. Because B ⊆ A this would means B ∩ V = ∅, a contradiction. 2

4.4 Compactness

Definition 4.19 A topological space X is called compact if for every collection Uωω∈Ω of open sets Uωwith the property X =

⋃ω∈Ω Uω there exist N and ω1, . . . , ωN ∈ Ω such that X =

⋃Nn=1 Uωn

.

A subset A of a topological space X is compact if A as a topological space with the subspace topologyis compact. This means that A ⊆ X is compact if if for every collection Uωω∈Ω of open sets Uω with the

property X ⊆⋃ω∈Ω Uω there exist N and ω1, . . . , ωN ∈ Ω such that X ⊆

⋃Nn=1 Uωn

.

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Theorem 4.20 (Heine-Borel) A subset of A ⊆ Rd is compact if and only if it is closed and bounded.

Examples 4.21

(1) A finite subset A of any topological space is compact.

(2) A topological space X with discrete topology is compact iff X is finite.

(3) Let X be a set with finite complement topology. Then any subset of X is compact.

Proposition 4.22 Let X and Y be topological spaces and g : X → Y be continuous. If X is compact, thenthe image A = f(X) is a compact subset of Y .

Corollary 4.23 Compactness is a topological property.

There is a certain relationship to closedness (as the Heine-Borel theorem suggests).

Proposition 4.24 Let A be a closed subset of a compact space X. Then A is compact.

Proposition 4.25 Let X be a Hausdorff space and let A be compact set. Then A is closed.

The assumption of Hausdorffness cannot be drop as the above examples show.In metric spaces one can give equivalent characterizations of compactness:

Theorem 4.26 Let X be a metric space and A ⊆ X. Then the following are equivalent:

(i) A is compact.

(ii) A is sequentially compact, i.e., for every sequence xn∞n=1 with xn ∈ A there exists a subsequencexnk

∞k=1 and X ∈ A such that xnk→ x.

(iii) A is limit-point compact, i.e., for every infinite subset B ⊂ A with have B′ 6= ∅.

(iv) A is closed and totally bounded.

Above, a subset A of a metric space is called totally bounded if for every ε > 0 there exist N andx1, . . . , xN ∈ X such that

A ⊆N⋃k=1

Bε(xk).

Here Bε(xk) are the ε-neighborhoods.

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