1. Department of Mathematics
MA4266 Topology
Lecture 2. Friday 15 Jan 2010
Wayne Lawton
Department of Mathematics
S ,

2. 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

3. Set Theory Theorem 1.3 Distributive Properties
Theorem 1.4 De Morgan’s Laws

4. Set Theory Theorem 1.5 Distributive Properties
Theorem 1.6 De Morgan’s 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

8. 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 ?
Question Let
and for
define
is finite. Is
an equivalence relation on

9. 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

11. Uncountable Sets Example 2.2.3
Theorem 2.2 The set of real numbers is uncountable.

12. 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.

13. Assignment 2 Read pages 14-52 in Chapters 1 and 2
Be prepared to solve any problems Tuesday.