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MA4266 Topology

Lecture 9. . MA4266 Topology. Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@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. Path Connected Spaces.

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MA4266 Topology

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  1. Lecture 9. MA4266 Topology Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@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

  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 are closed and Gluing Lemma If are continuous and and functions onto a space which satisfy then the function whenever is continuous. defined by Proof closed closed (in subspace ) and is a closed set is a closed set. Also is a closed set is a closed set. Hence is continuous.

  6. Joining Paths Definition The path product of path from to and path from to is defined by 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 ?

  7. Path Components Lemma If X is a space then the relation if there exists a path from to is an equivalence relation. Definition A path component of a space is a path connected subset which is not a proper subset of any path connected subset of Question 1. How are path components related to Lemma If X is an open subset of then 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 Definition A space is locally connected at a point if every open set containing contains a connected open set which contains A space is locally connected if it is locally connected at each point. Consider the Broom subspace Question 1 Is locally connected at Question 2. Is the Broom space locally connected ?

  9. Characterization Theorem 5.15 A space is locally connected at a point iif it has a local basis at consisting of connected sets and is locally connected iff it has a basis consisting of connected sets. Theorem 5.16 A space is locally connected iff for every open subset every component of is open. Proof  Let be a component of an open Then for every there exists an open connected Since is the largest with connected subset of containing then Then is open.  Left as an exercise.

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

  11. http://en.wikipedia.org/wiki/Connected_space http://en.wikipedia.org/wiki/Connectedness Web Links 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 Prepare for Friday’s Tutorial Read pages 147-157 Exercise 5.5 problems 1, 3, 10, 11 Exercise 5.6 problems 6, 12

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