Introduction to Topology

·       Theory of Sets

Topology is one of those branches of mathematics, where set theory is used extensively. As a matter of fact, set theory is the language of topology. In this course, we describe those concepts and results in the theory of sets, which are pre-requisite for the subject matter contained in the rest of the book. Accordingly, the contents of this course will be obtained so as to meet the requirements of set topology.

·       Sets and Subsets

We assume the reader is familiar with the term “set” and “elements” or “members” of a set. The relationship between a set S and any one of its elements, say x, is expressed symbolically by

x ∈ S

And read as

“x belongs to S OR x is an element (member or point) of S”

x ∉ S

it means that

x is not an element of S.

The set of which the elements are a, b, c, . . .  will be denoted by

{a, b, c, . . .}

The elements of set may themselves be sets. In topology collections of sets occur quite frequently. Although most of the concepts contained in this and the next session should be familiar to the reader, yet in order to make discussion on sets self-contained, we shall briefly review them.