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