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Math Help - Proving the compactness and connectedness of a subset

  1. #1
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    Proving the compactness and connectedness of a subset



    Any hint on how to approach this problem?

    Thanks as always
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by 6DOM View Post


    Any hint on how to approach this problem?

    Thanks as always
    Let g:[a,b]\longrightarrow\Gamma_f be g(x)=(x,f(x)). Let \{C_i\} be an open cover of \Gamma_f. The continuity of g implies that g^{-1}(C_i) is open in [a,b] (this is a theorem) and \{g^{-1}(C_i)\} is an open cover of [a,b].

    Spoiler:
    Since [a,b] is compact, we know that there are finitely many C_i such that [a,b]\subset\bigcup_{i=1}^N g^{-1}(C_i). Because g(g^{-1}(C_i))\subseteq C_i, we know that \Gamma_f\subset\bigcup_{i=1}^N C_i. So \Gamma_f is compact. \square


    You want to approach connectedness similarly. Assume that \Gamma_f\subset U\cup V where U and V are open sets such that cl(U)\cap V=U\cap cl(V)=\emptyset and try to derive a contradiction. (Your contradiction will be that [a,b] is not connected.)
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