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
The open sets in
are the balls D™E = {x G
: \\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
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
is a base for some topology if
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