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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

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

slide2

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

slide3

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 ?

slide4

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.

slide5

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.

slide6

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 ?

slide7

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.

slide8

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 ?

slide9

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.

slide10

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.

web links

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

assignment 9

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