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Math Help - compactness

  1. #1
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    compactness

    What are the examples of open sets that are compact?

    Is the entire space X compact?
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  2. #2
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    Quote Originally Posted by felixmcgrady View Post
    What are the examples of open sets that are compact?

    Is the entire space X compact?
    It depends on what topology you give on the space X.

    For example, if you give a discrete topology on a finite set X, then every open set is compact.

    Another example is that every open set is compact in the finite complement topology on \mathbb{Re}.

    Meanwhile, if you give a usual topology on \mathbb{Re}, then an open compact set is an empty set only.

    If you give a subspace topology on A with respect to the usual topology on \mathbb{Re}, where A is a finite subset of \mathbb{Re}, then every open set in the topological space A is compact.
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