Transcript MA4266_Lect9 - Department of Mathematics

MA4266 Topology
Lecture 9.
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, B  X are closed and X  A  B
g
:
B

Y
are continuous
f
:
A

Y
and
and
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
C  Y closed  h (C )  x  X : h( x)  C
 x  A : f ( x)  C x  B : g ( x)  C  f 1 (C )  g 1 (C )
 f 1 (C ) closed (in subspace A ) and A is a closed set
1
1
g
(C ) is a closed set
 f (C ) is a closed set. Also
 h 1 (C ) is a closed set. Hence h is continuous.
1
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.

xC
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 Friday’s Tutorial
Read pages 147-157
Exercise 5.5 problems 1, 3, 10, 11
Exercise 5.6 problems 6, 12