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Math Help - quotient space

  1. #1
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    quotient space

    Let X be a topological space and A a subset of X . On X {0, 1} define the
    partition composed of the pairs {(a, 0), (a, 1)} for a ∈ A , and of the singletons {(x, i)} if x ∈ X \ A and i ∈ {0, 1}.
    Let R be the equivalence relation defined by this partition, let Y be the quotient space
    [X {0, 1}] /R and let p : X {0, 1} → Y be the quotient map.

    (1) Prove that there exists a continuous map f : Y → X such that f ◦ p(x, i) = x for every x ∈ X and i ∈ {0, 1} .
    Prove that Y is Hausdorff if and only if X is Hausdorff and A is a closed subset of X .

    (2) Consider the above construction for X = [0, 1] and A an arbitrary subset of [0, 1].
    Prove that Y is compact. Prove that K = p(X {0}) and L = p(X {1}) are compact, and that K ∩ L is homeomorphic to A .

    Any input is appreciated!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by raulk View Post
    Let X be a topological space and A a subset of X . On X {0, 1} define the
    partition composed of the pairs {(a, 0), (a, 1)} for a ∈ A , and of the singletons {(x, i)} if x ∈ X \ A and i ∈ {0, 1}.
    Let R be the equivalence relation defined by this partition, let Y be the quotient space
    [X {0, 1}] /R and let p : X {0, 1} → Y be the quotient map.

    (1) Prove that there exists a continuous map f : Y → X such that f ◦ p(x, i) = x for every x ∈ X and i ∈ {0, 1} .
    What do you think?

    Prove that Y is Hausdorff if and only if X is Hausdorff and A is a closed subset of X .
    c
    The key point to notice is that the restriction X-A)\times\{0,1\}\to Y" alt="p^{*}X-A)\times\{0,1\}\to Y" /> is an injection, and thus if U\subseteq (X-A)\times\{0,1\} is open then so is p(U) since U=p^{-1}(p(U))

    (2) Consider the above construction for X = [0, 1] and A an arbitrary subset of [0, 1].
    Prove that Y is compact.
    Uhh... X is compact and evidently X\times\{0,1\} is compact and Y=p\left(X\times\{0,1\}\right) which is continuous?

    Prove that K = p(X {0}) and L = p(X {1}) are compact, and that K ∩ L is homeomorphic to A .
    X\times\{0,1\} is compact and both x\times\{0\} and X\times\{1\} are closed subspaces and so automatically compact. And since K,L are the continuous images of compact spaces they are trivially compact.

    Think about the other part. If x\in X-A then p((x,1))=\{(x,1)\}\notin p\left(X-A\times\{0\}\right) and so you can think of K\cap L as being A\times\{0,1\} but treating (a,0) and (a,1) as the same point. Doesn't this sound eerily familiar to A\times\{0\} which is naturally homeomorphic to A?
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