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Math Help - Compactness of topological space

  1. #1
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    Compactness of topological space

    Hey.

    My assignment says the following:

    Let X be a topolical space with  \tau as its topology. Let  \infty be a point not in X. Let X^* = X \cup \{ \infty \} .
    Let  \tau^* = \tau \cup \{ U \in X | X^* \setminus U is a closed, compact subset of X \}.

    (1) Prove that \tau^* is a topology on X^*. (I have already done this)

    (2) SHow that (X^* , \tau^* ) is compact.

    I dont know how to show number 2? Could anyone give me a hint or some advice on how to approach this?

    Thanks a million.

    /Morten
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  2. #2
    Super Member girdav's Avatar
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    Re: Compactness of topological space

    Take an open over of X^*, say (O_i)_{i\in I}. For some i_0, we have \alpha\in O_{i_0}, hence necessarily O_{i_0} is of the form X^*\setminus U_{i_0} where U_{i_0} is a closed compact subset of X. U_{i_0} is covered by (O_i)_{i\in I} hence you can extract a finite subcover and you are done (just add O_{i_0}).
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