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Thread: Compactness of topological space

  1. #1
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    Compactness of topological space

    Hey.

    My assignment says the following:

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

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

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