Transcript MA4266 Topology Lecture 4. Wayne Lawton

MA4266 Topology
Lecture 4.
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
Continuity
Thm 3.13 Let
A function
( X i , di ), i  1, 2
f : X1  X 2
be metric spaces.
is continuous when
any of the following equivalent conditions hold:
1. definition on page 75,
2. sequence ---,
3. open sets ---,
4. closed sets ---.
Equivalence
Definition Let
A function
( X i , di ), i  1, 2
f : X1  X 2
be metric spaces.
is
1. an isometry if ---,
2. a homeomorphism if --- .
Examples
Definition Uniform continuity and uniform equivalence.
New Spaces from Old
Definition Subspaces ---
Definition Product metric space --- .
Hilbert Space
 ( N ), N  1,2,3,...
2
Complete Metric Spaces
Definition Cauchy Sequences ---
Definition Complete Metric Space --- ,
Thm 3.16 A subspace (A,d) of a complete
metric space (X,d) is complete iff A is closed.
Baire Category
Definition Nowhere dense --Definition 1st category and 2nd category space
Thm 3.17 Baire Category Theorem. Every
complete metric space is of second category.
Definition Contractive functions --Thm 3.18 Baire Category Theorem. Every
complete metric space is of second category.
Assignment 4
Read pages 82-97 in Chapter 3
Solve all problems in Exercises
Written Homework Problems Due Tues 1 February
1. Do problem 4 on page 77
2. Do problem 6 on page 82
3. Do problem 5 on page 86
4. Do problem 7 on page 95
TEST 1. Tuesday 15 February 2011