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
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