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Unformatted text preview: 1.2. MANIFOLDS 7 Definition 1.2.1 A general topological space is a set M together with a collection of subsets of M, called open sets , that satisfy the following properties 1. M and are open, 2. the intersection of any finite number of open sets is open, 3. the union of any collection of open sets is open. Such a collection of open sets is called a topology of M. A subset of M is closed if its complement M \ F is open. The closure S of a subset S M of a topological space M is the intersec- tion of all closed sets that contain S ; it is equal to S = S S . A subset S M of a topological space is dense in M if S = M , that is, every non-empty subset of M contains an element of S . A topological space is called separable if it contains a countable dense subset. A topology on M naturally induces a topology on any subset of M . Let A M be a subset of M . Then the induced topology on A is defined as follows. A subset V A is defined to be open subset of...
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