Chapter 1

Topological spaces and

operations with them

Each lecturer starting a course in topology feels obliged to s

topology studies those properties of geometric objects that do

pend on distances, curvatures, and other metric values, i.e., pro

invariant with respect to continuous deformations of the objec

shall explain what this means by the end of this chapter.

The elegant geometric construction and ideas that were pr

in the Foreword will be explained later, and we start with

definitions.

Topology studies topological spaces and their continuous

1.1. Topological spaces and homeomorphisms

Definition. A topological space is a set X endowed with a topo

structure (a topology) r. Here a topological structure r is some

of subsets of X: r C 2 X , whose elements are called open. The

of open sets should satisfy the following properties:

(1) the union of any set of open subsets is an open subse

(2) the intersection of a finite collection of open sets is a

set;

(3) the empty set 0 and the whole set X are open.

http://dx.doi.org/10.1090/stml/01