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We start with def a topological space x t is a set x

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Unformatted text preview: ach. We start with: Def.: A topological space (X, T) is a set X together with a topology T on X. A topology on a set X is a collection of subsets of X (that is, T P(X)) satisfying: 1. If G1, G2 T, then G1 G2 T. 2. If {G | J} is any collection of sets in T, then G T. J 3. T, and X T. 4 The sets G T are called open sets in X. A subset F X whose complement is open is called a closed set in X. If A is any subset of a topological space X, then the interior of A, denoted by A, is the union of all open sets contained in A. The closure of A, denoted by A, is the intersection of all closed sets containing A. If x X, then a neighborhood of x is any subset A X with x A. If (X, T ) is a topological space, and A is a subset of X, then the induced or subspace topology TA on A is given by TA = {G A | G T }. It is easy to check that TA actually is a topology on A. With this topology, A is called a subspace of X. 5 Suppose X and Y are topological spaces, and f : X Y . Recall that if V Y , we use the notation f -1(V ) = {x X | f (x) V }. We then have the definit...
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