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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 nonempty 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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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Logic, Geometry, Sets

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