1. MA4266 Topology Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sgmatwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1 Lecture 9.

2. 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 that connects a to b

3. 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. Challenging Example Is it pathwise connected ?

4. 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.

5. Joining Paths Gluing Lemma If are closed and and are continuous functions onto a spacewhich satisfy whenever then the function defined by is continuous. Proof closed and closed (in subspace) andis a closed set is a closed set. Alsois a closed set is a closed set. Henceis continuous.

6. Joining Paths Definition The path product of pathtofromto is defined by and path Question 1. The path product is a path from ? to ? from Question 2. Why is the path product continuous ? Question 3. Is the path product associative ?

7. Path Components Definition A path component of a spaceis a path connected subset which is not a proper subset of any is an equivalence relation. if there exists a path fromto Lemma If X is a space then the relation path connected subset of Question 1. How are path components related to Lemma If X is an open subset ofthen every path component is open. Corollary Under this hypothesis every p.c. is also closed. Theorem 5.12 Every open, connected subset of is path connected.

8. Local Connectedness containingcontains a connected open set is locally connected at a point if every open set Definition A space Consider the Broom Question 1 Islocally connected at Question 2. Is the Broom space locally connected ? subspace A spaceis locally connected if it is locally connected at each point. which contains

9. Characterization connected sets. is locally connected at a pointTheorem 5.15 A space is locally connected iffTheorem 5.16 A space and is locally connected iff it has a basis consisting of iif it has a local basis atconsisting of connected sets for every open subsetevery component of is open. Proof  Letbe a component of an open Then for everythere exists an open connected with Sinceis the largest connected subset ofcontainingthen Thenis open.  Left as an exercise.

10. Local Path Connectedness containingcontains a path connected open set is locally path connected at a point open set Definitionif every A spaceis locally path connected if it is locally path connected at each point. which contains Theorem 5.17is locally path connected at a point iif it has a local basis atconsisting of path connected sets and is locally path connected iff it has a basis consisting of path connected sets. Theorem 5.18is locally path connected iff for every openevery path component ofis open. Theorem 5.19 Conn. & local path conn.  path conn.

11. Web Links http://en.wikipedia.org/wiki/Connected_space http://en.wikipedia.org/wiki/Connectedness http://www.aiml.net/volumes/volume7/Kontchakov-PrattHartmann-Wolter- Zakharyaschev.pdf http://www.cs.colorado.edu/~lizb/topology.html http://people.physics.anu.edu.au/~vbr110/papers/nonlinearity.html

12. Assignment 9 Read pages 147-157 Exercise 5.5 problems 1, 3, 10, 11 Exercise 5.6 problems 6, 12 Prepare for Friday’s Tutorial