Department of Mathematics MA4266 Topology Lecture 2. Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1

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

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

Set Theory Theorem 1.5 Distributive Properties Theorem 1.6 De Morgan’s Laws

Cartesian Products Definition For an indexed collection of sets If

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?

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

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

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.

Finite and Infinite Sets Finite versus infinite sets. Equipotent sets. Countable sets. Example 2.2.2

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

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.

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