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Unformatted text preview: 1st December 2004 Munkres Â§ 13 Ex. 13.1 (Morten Poulsen). Let ( X, T ) be a topological space and A â X . The following are equivalent: (i) A â T . (ii) â x â A â U x â T : x â U x â A . Proof. (i) â (ii): If x â A then x â A â A and A â T . (ii) â (i): A = S x â A U x , hence A â T . Ex. 13.4 (Morten Poulsen). Note that every collection of topologies on a set X is itself a set: A topologi is a subset of P ( X ), i.e. an element of P ( P ( X )), hence a collection of topologies is a subset of P ( P ( X )), i.e. a set. Let {T Î± } be a nonempty set of topologies on the set X . (a) . Since every T Î± is a topology on X it is clear that the intersection T T Î± is a topology on X . The union S T Î± is in general not a topology on X : Let X = { a, b, c } . It is straightforward to check that T 1 = { X, â
, { a } , { a, b }} and T 2 = { X, â
, { c } , { b, c }} are topologies on X . But T 1 âȘT 2 is not a topology on X , since { a, b } â© { b, c } = { b } / â T 1 âȘ T 2 . (b) . The intersection of all topologies that are finer than all T Î± is clearly the smallest topology containing all T Î± . The intersection of all T Î± is clearly the largest topology that is contained in all T Î± . (c) . The topology T 3 = T 1 â©T 2 = { X, â
, { a }} is the largest topology on X contained in T 1 and T 2 ....
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This note was uploaded on 01/12/2011 for the course MATH 110 taught by Professor Brown during the Fall '08 term at Arizona Western College.
 Fall '08
 Brown
 Logic, Topology

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