MA4266 Topology

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MA4266 Topology
Lecture 1.

• 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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Administrative

• MA4266 Module Title TOPOLOGY Semester 2,
  2010/2011
• Modular Credits 4 Faculty Science Department
  Mathematics
• Teaching Staff ASSOC PROF Lawton, Wayne Michael
  matwml_at_nus.edu.sg
• http//www.math.nus.edu.sg/matwml/courses/ my
  personal website which contains mountains of
  materials
• AIMS OBJECTIVES The objective of this module is
  to give a thorough introduction to the topics of
  point-set topology with applications to analysis
  and
• geometry. Major topics topological spaces,
  continuous maps, bases, subbases, homeomorphisms,
  subspaces, sum and product topologies, quotient
  spaces
• and identification maps, orbit spaces, separation
  axioms, compact spaces, Tychonoff's theorem,
  Heine-Borel theorem, compactness in metric space,
  
• sequential compactness, connected and
  path-connected spaces, components, locally
  compact spaces, function spaces and the
  compact-open topology.
• PREREQUISITES MA3209 Mathematical Analysis III
• TEACHING MODES
• Lectures, questions, discussions, tutorial
  problem solving and presentation by students and
  subsequent discussion encouraged by questions
  from the
• lecturer, assigned readings covering most chapters
  from the textbook Principles of Topology by Fred
  Croom and supplementary materials taken from
• Various sources,  two tests and a final
  examination, written homework that is collected
  and graded and handed back to students.
• SCHEDULE
• Final Examination
• LECTURE Class SL1
• TUESDAY From 1600 hrs to 1800 hrs in S16-0430,
• Week(s) EVERY WEEK.
• FRIDAY From 1600 hrs to 1800 hrs in S16-0430,

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Textbook
Principles of Topology by Fred H Croom, Thompson,
Singapore, 1989.
Available in the Science COOP Bookstore at a
significantly reduced student price
The use of this textbook is compulsory
because you are expected to read most of it and
work out solutions to selected problems located
at the ends of each of the 8 chapters.
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Contents of Textbook

1. Introduction
2. The Line and the Plane
3. Metric Spaces
4. Topological Spaces
5. Connectedness
6. Compactness
7. Product and Quotient Spaces
8. Separation Properties and Metrization
9. The Fundamental Group

My aim to is cover all of the material in the
textbook
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What is Topology ?
Greek
position or location
in the sense of properties that are
NOT destroyed by continuous transformations
bending, shrinking, stretching and twisting
BUT are destroyed by discontinuous
transformations
cutting, tearing, and puncturing
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Example 1.1.1
Shrinking in the vertical direction
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Non Morphable Example
Involution in a circle
on
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Topological Equivalence
and
Between Geometric Objects
means there exists is a continuous bijection
is also continuous.
whose inverse
Which pairs below of geom. obj. are top. equiv.?
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Who Needs Topology ?
Theorem 1.1 The Intermediate Value Theorem.
is continuous and there exists
If
and
such that
such that
then there exists
Corollary If
is defined by a polynomial
having odd degree then
has a real root.
In what other areas, aside from calculus and
algebra, are existence theorems important?
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Origins of Topology
In 1676 Leibnitz used the term geometria
situs, Latin for geometry of position, to
designate what he predicted to be the
development of a new type of geometry similar to
modern day topology
http//en.wikipedia.org/wiki/Gottfried_Leibniz
Example 1.2.1 In 1736 Euler solved The Königsberg
bridges problem, this inventing graph theory
http//en.wikipedia.org/wiki/Seven_Bridges_of_KC3
B6nigsberg
Psychologists hypothesize that the human brain is
topologically wired !
http//www.jstor.org/pss/748762
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Assignment 1
Read Preface and pages 1-14 in Chapter 1
Do Problems 1-6 on pages 13-14 and prepare to
solve on the board in class for Friday