MA4266 Topology

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MA4266 Topology
Lecture 4.

• Wayne Lawton
• Department of Mathematics
• S17-08-17, 65162749 matwml_at_nus.edu.sg
• http//www.math.nus.edu.sg/matwml/
• http//arxiv.org/find/math/1/auLawton_W/0/1/0/al
  l/0/1

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Continuity
Thm 3.13 Let
be metric spaces.
is continuous when
A function
any of the following equivalent conditions hold
1. definition on page 75,
2. sequence ---,
3. open sets ---,
4. closed sets ---.
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Equivalence
Definition Let
be metric spaces.
is
A function
1. an isometry if ---,
2. a homeomorphism if --- .
Examples
Definition Uniform continuity and uniform
equivalence.
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New Spaces from Old
Definition Subspaces ---
Definition Product metric space --- .
Hilbert Space
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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.
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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.
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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