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MA4266 Topology Lecture 9. Tuesday 23 Feb 2010 Wayne Lawton Department of Mathematics S17-08-17, 65162749 [email protected] http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1 Path Connected Spaces Definition A topological space X is pathwise connected if for every a and b in X there exists a path p in X p : [0,1] X that connects a to b p(0) a, p(1) b Examples Definition A subset C of Euclidean space (of any dimension) is convex if the line segment connecting any two points in C lies within C. p(t ) (1 t )a tb a t (b a), t [0,1] Challenging Example 2 R \Q Is it pathwise connected ? 2 Relation With Connectedness Theorem 5.11 Every pathwise connected space is connected. Example 5.5.3 The Topologists Sine Curve is connected but not pathwise connected. Example 5.5.4 The space below is connected but not pathwise connected. Joining Paths Gluing Lemma If A and B are subspaces of X A B and f : A Y and g : B Y are continuous functions onto a space Y which satisfy f ( x) g ( x) whenever x A B then the function h : X Y f ( x), x A is continuous. defined by h( x) g ( x), x B Proof If C Y is closed then h (C ) x X : h( x) C 1 x A : f ( x) C x B : g ( x) C f 1 ( A) g 1 ( B) Joining Paths Definition The path product p p : [0,1] X 1 2 of path p1 from a to b and path p2 from b to c is defined by p1 (2 t ), t [0, 12 ] p1 p2 (t ) 1 p2 (2 t 1), t [ 2 ,1] Question 1. The path product is a path from ? to ? Question 2. Why is the path product continuous ? Question 3. Is the path product associative ? Path Components Lemma If X is a space then the relation a b if there exists a path from a to b is an equivalence relation. Definition A path component of a space X is a path connected subset which is not a proper subset of any path connected subset of X . Question 1. How are path components related to ? Lemma If X is an open subset of path component is open. R n then every Corollary Under this hypothesis every p.c. is also closed. Theorem 5.12 Every open, connected subset of is path connected. R n Local Connectedness Definition A space X is locally connected at a point p if every open set U containing p contains a connected open set V which contains p. A space X is locally connected if it is locally connected at each point. Consider the Broom subspace X R 2 Question 1 Is 0 X locally connected at 0? Question 2. Is the Broom space locally connected ? Characterization Theorem 5.15 A space X is locally connected at a point p iif it has a local basis at p consisting of connected sets and is locally connected iff it has a basis consisting of connected sets. Theorem 5.16 A space X is locally connected iff for every open subset O X every component of O is open. Proof Let C be a component of an open O X . Then for every x C there exists an open connected U with x U O. Since C is the largest x x connected subset of O containing x then U x C. Then C U is open. Left as an exercise. xC x Local Path Connectedness Definition X is locally path connected at a point p if every open set U containing p contains a path connected open set V which contains p. A space X is locally path connected if it is locally path connected at each point. Theorem 5.17 X is locally path connected at a point iif it has a local basis at p p consisting of path connected sets and is locally path connected iff it has a basis consisting of path connected sets. X is locally path connected iff for every O X every path component of O is open. Theorem 5.18 open Theorem 5.19 Conn. & local path conn. path conn. Web Links http://en.wikipedia.org/wiki/Connected_space http://en.wikipedia.org/wiki/Connectedness http://www.aiml.net/volumes/volume7/Kontchakov-PrattHartmann-WolterZakharyaschev.pdf http://www.cs.colorado.edu/~lizb/topology.html http://people.physics.anu.edu.au/~vbr110/papers/nonlinearity.html Assignment 9 Prepare for Thursday’s Tutorial Read pages 147-157 Exercise 5.5 problems 1, 3, 10, 11 Exercise 5.6 problems 6, 12