TOPOLOGY I M 367K
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Date Created: 09/06/15
Chapter 4 Maps Between Topological Spaces Often in mathematics once we have de ned some sort of a mathematical structure we then turn our attention to describing functions that acknowl edge that structure So now we turn our attention to functions between topological spaces where our goal is to decide what properties of the func tions will describe a relationship between the topologies of the two spaces involved Speci cally our rst task is to decide what functions between topological spaces we want to call continuous Also we want to de ne what it means to say that two topological spaces are the same 41 Continuity Recall our analysis of the de nition of continuity of a real valued function on the reals By re phrasing it in terms of open sets we are able to create a de nition of continuity that is meaningful for functions between any two topological spaces De nition continuous function Let X and Y be topological spaces A function f X a Y is a continuous function if and only if for every open set U in Y f 1U is open in X In this course we will use the terms map and continuous function synonymously Theorem 41 Let f X a Y be a function Then the following are equiva lent 1 f is continuous 2 for every closed set K in Y f 1K is closed in X 33 34 CHAPTER 4 MAPS BETWEEN TOPOLOGICAL SPACES 3 ifp is a limit point of A in X then fp belongs to fA To verify that our de nition of continuity is a good one let s verify that in the context of metric spaces the traditional 6 6 de nition of continuity is equivalent to the inverse images of open sets are open de nition Theorem 42 If X and Y are metric spaces with metrics dX and dy re spectively then a function f X a Y is continuous if and only if for each point z in X and e gt 0 there is a 6 gt 0 such that for each y E X with dXxy lt 6 then dyfz lt 8 When metric spaces are involved continuity can be described in terms of convergence Theorem 43 Let X be a metric space and Y be a topological space Then a function f X a Y is continuous if and only if for each convergent sequence m a m x converges to For functions between metric spaces there is a stronger concept than continuity De nition uniformly continuous A function 1 from a metric space X dX to a metric space Y dy is uniformly continuous if and only if for each 6 gt 0 there is a 6 gt 0 such that for every m y E X if dXmy lt 6 then dyf907 fy lt 6 Erercise 44 Give an example of a continuous function from R1 to R1 with the standard topology that is not uniformly continuous Theorem 45 Let f X a Y be a continuous function from a compact metric space to a metric space Y Then 1 is uniformly continuous Continuous functions preserve some of the topological properties we have studied Theorem 46 Let X be a compact respectively Lindelof countably com pact space and let 1 X a Y be a continuous function that is onto Then Y is compact respectively Lindelof countably compact Theorem 47 Let X be a separable space and let 1 X a Y be a continuous onto map Then Y is separable There is a relationship between normality of a space X and the existence of some continuous functions from X into 0 1 with the standard topology That important relationship is captured in a theorems known as Urysohn s Lemma and the Tietze Extension Theorem The next lemma is used in the proof of Urysohn s Lemma 4 2 HOME OMORPHISMS 35 Lemma 48 Let A and B be disjoint closed sets in a normal space X Then for each diadic rational r E 0 1 r is a diadic rational if and only if it is of the form qQk where q k are integers there exists an open set U such that AQUO B X7U1andforrlts77 Us Urysohn Lemma 49 A space X is normal if and only if for each pair of disjoint closed sets A and B in X there exists a continuous function f X a 01 such that A Q f 10 and B Q f 11 Understanding the relationship between continuous functions and nested open sets allows us to prove the Tietze Extension Theorem below Other proofs can be created that apply the statement of Urysohn s Lemma repeat edly to get a sequence of functions that converge to the desired function But the proofs of the Tietze Extension Theorems are still dif cult Tietze Extension Theorem 410 A space X is normal if and only if every continuous function 1 from a closed set A in X into 0 1 can be extended to a continuous function F X a 0 1 F extends 1 means for each point m in A Tietze Extension Theorem 411 A space X is normal if and only if every continuous function 1 from a closed set A in X into 0 1 can be extended to a continuous function F X a 0 1 F extends 1 means for each point m in A De nition closed and open functions A continuous function f X a Y is closed if and only if for every closed set A in X fA is closed in Y A continuous function f X a Y is open if and only if for every open set U in X fU is open in Y Theorem 412 Let X be compact and Y be Hausdorff Then any continuous function f X a Y is closed 42 Homeomorphisms We now turn to the question of when two topological spaces are the same De nition homeomorphism A function f X gt Y is a homeomorphism if and only if f is continuous 171 and onto and f 1 Y a X is also continuous De nition homeomorphic spaces X and Y two topological spaces are said to be homeomorphic if and only if there exists a homeomorphism f X gt Y 36 CHAPTER 4 MAPS BETWEEN TOPOLOGICAL SPACES Theorem 413 For a continuous function f X a Y the following are equivalent a f is a homeomorphism h f is 171 onto and closed c f is 171 onto and open Theorem 414 For points a lt b in R1 with the standard topology the interval a b is homeomorphic to R1 Theorem 415 Suppose f X a Y is a 171 and onto continuous function X is compact and Y is Hausdorff Then 1 is a homeomorphism Theorem 416 Let f X a Y be a function Suppose X A U B where A and B are closed subsets ofX If f l A is continuous and f l B is continuous then f is continuous The following statement cannot be proven without more rigorous def initions In what sense could it be made rigorous ls there a reasonable de nition of a topological property Metatheorem 417 If X and Y are topological spaces and f X a Y is a homeomorphism then X and Y are the same as topological spaces 239e any topological property of the space X is also a topological property of the space Y
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