S13 - 1st December 2004 Munkres  13 Ex 13.1(Morten Poulsen Let X T be a topological space and A ⊂ X The following are equivalent(i A ∈ T(ii

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Unformatted text preview: 1st December 2004 Munkres § 13 Ex. 13.1 (Morten Poulsen). Let ( X, T ) be a topological space and A ⊂ X . The following are equivalent: (i) A ∈ T . (ii) ∀ x ∈ A ∃ U x ∈ T : x ∈ U x ⊂ A . Proof. (i) ⇒ (ii): If x ∈ A then x ∈ A ⊂ A and A ∈ T . (ii) ⇒ (i): A = S x ∈ A U x , hence A ∈ 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 α } be a nonempty set of topologies on the set X . (a) . Since every T α is a topology on X it is clear that the intersection T T α is a topology on X . The union S T α is in general not a topology on X : Let X = { a, b, c } . It is straightforward to check that T 1 = { X, ∅ , { a } , { a, b }} and T 2 = { X, ∅ , { c } , { b, c }} are topologies on X . But T 1 âˆȘT 2 is not a topology on X , since { a, b } ∩ { b, c } = { b } / ∈ T 1 âˆȘ T 2 . (b) . The intersection of all topologies that are finer than all T α is clearly the smallest topology containing all T α . The intersection of all T α is clearly the largest topology that is contained in all T α . (c) . The topology T 3 = T 1 ∩T 2 = { X, ∅ , { 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.

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S13 - 1st December 2004 Munkres  13 Ex 13.1(Morten Poulsen Let X T be a topological space and A ⊂ X The following are equivalent(i A ∈ T(ii

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