MA4266 Topology

1
MA4266 Topology
Lecture 2.

• 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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Indexed Sets
Definition (p. 13)
a set and for each
there is a set
The collection
is said to be indexed by
Example Let
Question What is
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Set Theory
Theorem 1.3 Distributive Properties
Theorem 1.4 De Morgans Laws
4
Set Theory
Theorem 1.5 Distributive Properties
Theorem 1.6 De Morgans Laws
5
Cartesian Products
Definition For an indexed collection of sets
If
6
Questions
and
If
what is the relationship between
and
Show that a tangent vector field on a sphere is
an element of a certain Cartesian product.
Is every element of the Cartesian product a
vector field?
7
Relations and Functions
Question What is the difference between the range
and the image of a function ?
Question Is every function a relation ? Is
every relation a function ?
Question Does every relation have an inverse
relation ?
Question Consider a function
When does it have a left inveres ? When does it
have a right inverse ?
If
describe the restriction
Describe a function
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Equivalence Relations
Question What is meant by reflexive, symmetric,
and transitive ?
Question What is an example of an equivalence
relation on Z that has 5 equivalence classes ?
and for
Question Let
is finite.
define
an equivalence relation on
Is
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Upper and Lower Bounds
Intervals give examples of four types of
bounded intervals and four types of unbounded
intervals.
What are their least upper bounds (supremums)
and greatest lower bounds (infinums) ?
Explain the Least Upper Bound Property for
Show that it is equivalent to the GLBP.
Theorem 2.1 Between every two real numbers there
is a rational number.
10
Finite and Infinite Sets
Finite versus infinite sets.
Equipotent sets.
Countable sets.
Example 2.2.2
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Uncountable Sets
Example 2.2.3
Theorem 2.2 The set of real numbers is
uncountable.
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Open and Closed Sets
Distance
Definition Open ?
Definition Closed ?
Accumulation Point ?
Theorem 2.10 A subset of R is closed iff it
contains all of its accumulation points.
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Assignment 2
Read pages 14-52 in Chapters 1 and 2
Be prepared to solve any problems Tuesday.