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Math Help - Proofs for Topology

  1. #1
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    Proofs for Topology

    Hey I don't know if anyone can help but here goes:I need to proof these 3 statements:

    Let X have a discrete topolgy and Y be an arbitrary topological space. Show that every funtion f:X maps to Y is continuous.

    Let y have the trivial topology and X be an arbitrary topological space. Show that every funtion f: X map to Y is coninuous.

    Let X & Y be topological spaces. A function f:X maps to Y is continuous if and only in the inverse (C) is closed in X for every closed set C is a subset of Y.
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  2. #2
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    Quote Originally Posted by reagan3nc View Post
    Let X have a discrete topolgy and Y be an arbitrary topological space. Show that every funtion f:X maps to Y is continuous.
    Let x_0 \in X we need to prove that for any \epsilon > 0 there exists \delta > 0 such that if d(x,x_0)<\delta \mbox{ and }x\in X\implies d'(f(x),f(x_0))<\epsilon. Pick \delta = 1/2 then for d(x,x_0)<\delta \mbox{ and }x\in X to be true we require that x=x_0 since it is the discrete metric. But then clearly d'(f(x),f(x_0))=0<\epsilon.

    Let y have the trivial topology and X be an arbitrary topological space. Show that every funtion f: X map to Y is coninuous.
    A function f:X\mapsto Y from metric spaces (X,d) and (Y,d') is continous if and only if the inverse-image of every open set is open. Since Y is a trivial topology it means the only open sets are \emptyset and Y. But f^{-1} [\emptyset] = \emptyset and f^{-1}[Y] = X. But \emptyset and X are open. Thus, f was a continous function.

    Let X & Y be topological spaces. A function f:X maps to Y is continuous if and only in the inverse (C) is closed in X for every closed set C is a subset of Y.
    Let C be a closed subset of Y. Consider \bar C (the complement). This set is open. And since f is continous it means f^{-1}[\bar C] = \widehat{f^{-1}[C]} is open. Thus, f^{-1}[C] is closed. The converse is similar.
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