Lawson topologyFrom Wikipedia, the free encyclopedia

Contents

1 Lawson topology 11.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Lexicographic order topology on the unit square 22.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Limit (mathematics) 33.1 Limit of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Limit as standard part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Convergence and xed point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 Topological net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Limit point 74.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Types of limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Some facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Lindelf space 105.1 Properties of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Properties of strongly Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3 Product of Lindelf spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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5.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 List of examples in general topology 126.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Local homeomorphism 147.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

8 Local property 168.1 Properties of a single space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.2 Properties of a pair of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.3 Properties of innite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.4 Properties of nite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.5 Properties of commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

9 Locally connected space 189.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.2 Denitions and rst examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9.2.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.4 Components and path components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.5 Quasicomponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

9.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.6 More on local connectedness versus weak local connectedness . . . . . . . . . . . . . . . . . . . . 229.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

10 Locally nite collection 2410.1 Examples and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10.1.1 Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410.1.2 Second countable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.3 Countably locally nite collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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10.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11 Locally nite space 2611.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Chapter 1

Lawson topology

In mathematics and theoretical computer science the Lawson topology, named after J. D. Lawson, is a topologyon partially ordered sets used in the study of domain theory. The lower topology on a poset P is generated by thesubbasis consisting of all complements of principal lters on P. The Lawson topology on P is the smallest commonrenement of the lower topology and the Scott topology on P.

1.1 Properties If P is a complete upper semilattice, the Lawson topology on P is always a complete T1 topology.

1.2 See also Formal ball

1.3 References G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott (2003), Continuous Latticesand Domains, Encyclopedia of Mathematics and its Applications, Cambridge University Press. ISBN 0-521-80338-1

1.4 External links "How Do Domains Model Topologies?, Pawel Waszkiewicz, Electronic Notes in Theoretical Computer Sci-ence 83 (2004)

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Chapter 2

Lexicographic order topology on the unitsquare

In general topology, the lexicographic ordering on the unit square is a topology on the unit square S, i.e. on theset of points (x,y) in the plane such that 0 x 1 and 0 y 1.[1]

2.1 ConstructionAs the name suggests, we use the lexicographical ordering on the square to dene a topology. Given two points inthe square, say (x,y) and (u,v), we say that (x,y) (u,v) if and only if either x < u or both x = u and y < v. Given thelexicographical ordering on the square, we use the order topology to dene the topology on S.

2.2 PropertiesThe order topology makes S into a completely normal Hausdor space. [1] It is an example of an order topology inwhich there are uncountably many pairwise-disjoint homeomorphic copies of the real line. Since the lexicographicalorder on S can be proven to be complete, then this topology makes S into a compact set. At the same time, S is notseparable, since the set of all points of the form (x,1/2) is discrete but is uncountable. Hence S is not metrizable(since any compact metric space is separable); however, it is rst countable. [1]

2.3 See also Long line

2.4 References[1] Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 73, ISBN 0-486-68735-X

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Chapter 3

Limit (mathematics)

This is an overview of the idea of a limit in mathematics. For specic uses of a limit, see Limit of a sequence andLimit of a function.

In mathematics, a limit is the value that a function or sequence approaches as the input or index approaches somevalue.[1] Limits are essential to calculus (and mathematical analysis in general) and are used to dene continuity,derivatives, and integrals.The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closelyrelated to limit and direct limit in category theory.In formulas, a limit is usually written as

limn!c f(n) = L

and is read as the limit of f of n as n approaches c equals L". Here lim indicates limit, and the fact that functionf(n) approaches the limit L as n approaches c is represented by the right arrow (), as in

f(n)! L :

3.1 Limit of a functionMain article: Limit of a function

Suppose f is a real-valued function and c is a real number. The expression

limx!c f(x) = L

means that f(x) can be made to be as close to L as desired by making x suciently close to c. In that case, the aboveequation can be read as the limit of f of x, as x approaches c, is L".Augustin-Louis Cauchy in 1821,[2] followed by Karl Weierstrass, formalized the denition of the limit of a functionas the above denition, which became known as the (, )-denition of limit in the 19th century. The denition uses (the lowercase Greek letter epsilon) to represent any small positive number, so that "f(x) becomes arbitrarily closeto L" means that f(x) eventually lies in the interval (L , L + ), which can also be written using the absolute valuesign as |f(x) L| < .[2] The phrase as x approaches c" then indicates that we refer to values of x whose distancefrom c is less than some positive number (the lower case Greek letter delta)that is, values of x within either (c , c) or (c, c + ), which can be expressed with 0 < |x c| < . The rst inequality means that the distance betweenx and c is greater than 0 and that x c, while the second indicates that x is within distance of c.[2]

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Note that the above denition of a limit is true even if f(c) L. Indeed, the function f need not even be dened at c.For example, if

f(x) =x2 1x 1

then f(1) is not dened (see division by zero), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches2:Thus, f(x) can be made arbitrarily close to the limit of 2 just by making x suciently close to 1.In other words, limx!1 x

21x1 = 2

This can also be calculated algebraically, as x21x1 =(x+1)(x1)

x1 = x+ 1 for all real numbers x 1.

Now since x + 1 is continuous in x at 1, we can now plug in 1 for x, thus limx!1 x21x1 = 1 + 1 = 2 .

In addition to limits at nite values, functions can also have limits at innity. For example, consider

f(x) =2x 1x

f(100) = 1.9900 f(1000) = 1.9990 f(10000) = 1.99990

As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 asone could wish just by picking x suciently large. In this case, the limit of f(x) as x approaches innity is 2. Inmathematical notation,

limx!1

2x 1x

= 2:

3.2 Limit of a sequenceMain article: Limit of a sequence

Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are approaching 1.8,the limit of the sequence.Formally, suppose a1, a2, ... is a sequence of real numbers. It can be stated that the real number L is the limit of thissequence, namely:

limn!1 an = L

which is read as

The limit of an as n approaches innity equals L"

to mean

For every real number > 0, there exists a natural number N such that for all n > N, we have |an L|