Basic Definitions

To start the book, we only need some basic notions of topology. We present

them below.

A topological space is a set X with a system r of distinguished subsets

having the following properties:

(i) the empty set and the entire set X belong to r;

(ii) the intersection of any finite set of elements of r belongs to r;

(iii) the union of an arbitrary family of elements of r belongs to r.

The sets belonging to r are said to be open. A neighborhood of a point

x G X is an arbitrary open set containing x. The sets with open comple-

ments are closed.

Let A be a subset in a topological space. Its closure A is defined as the

minimal closed set containing A, and its interior int A is the maximal open

set contained in A. The closure of A is the intersection of all closed sets

containing A, and the interior of A is the union of all open sets contained

in A.

The most important example of a topological space is the Euclidean

space

Rn.

The open sets in

W1

are the balls D™E = {x G

Rn

: \\x - a\\ e]

and all their unions.

A family r' C r is called a base for the topology r if any element of r is

a union of elements of

rr.

Exercise 1. Prove that a family r' C r is a base for the topology r if and

only if, for any point x and any neighborhood U of x, there exists a V G T'

such that x G V C U.

Exercise 2. Prove that a family of sets

rf

is a base for some topology if

1