01-top - Chapter I Topology Preliminaries Lecture 1 1....

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Unformatted text preview: Chapter I Topology Preliminaries Lecture 1 1. Review of basic topology concepts In this lecture we review some basic notions from topology, the main goal being to set up the language. Except for one result (Uryson Lemma) there will be no proofs. Definitions. A topology on a (non-empty) set X is a family T of subsets of X , which are called open sets, with the following properties: ( top 1 ): both the empty set and the total set X are open; ( top 2 ): an arbitrary union of open sets is open; ( top 3 ): a finite intersection of open sets is open. In this case the system ( X, T ) is called a topological space . If ( X, T ) is a topological space and x X is an element in X , a subset N X is called a neighborhood of x if there exists some open set D such that x D N . A collection N of neighborhoods of x is called a basic system of neighborhoods of x , if for any neighborhood M of x , there exists some neighborhood N in N such that x N M . A collection V of neighborhoods of x is called a fundamental system of neighbor- hoods of x if for any neighborhood M of x there exists a finite sequence V 1 ,V 2 ,...,V n of neighborhoods in V such that x V 1 V 2 V n M . A toplogy is said to have the Hausdorff property if: (h) for any x,y X with x 6 = y , there exist open sets U 3 x and V 3 y such that U V = . If ( X, T ) is a topological space, a subset F X will be called closed , if its complement X r F is open. The following properties are easily derived from the definition: ( c 1 ) both the empty set and the total set X are closed; ( c 2 ) an arbitrary intersection of closed sets is closed; ( c 3 ) a finite union of closed sets is closed. Using the above properties of open/closed sets, one can perform the following constructions. Let ( X, T ) be a topological space and A X be an arbitrary subset. Consider the set Int( A ) to be the union of all open sets D with D A and consider the set A to be the intersection of all closed sets F with F A . The set Int( A ) (sometimes denoted simply by A ) is called the interior of A , while the set A is called the closure of A . The properties of these constructions are summarized in the following: Proposition 1.1. Let ( X, T ) be a toplogical space, and let A be an arbitrary subset of X . 3 4 LECTURE 1 A. (Properties of the interior) (i) The set Int( A ) is open and Int( A ) A . (ii) If D is an open set such that D A , then D Int( A ) . (iii) x belongs to Int( A ) if and only if A is a neighborhood of x ....
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01-top - Chapter I Topology Preliminaries Lecture 1 1....

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