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Math Help - Definition of sigma-compact

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    Definition of sigma-compact

    I got two definitions of \sigma-compact: one is from Munkres' "Topology" page 289, ex. 10, as the image below;


    the other is from wikipedia, which says \sigma-compact means the space is the union of countably many compact subspaces.
    I can deduce from Munkres' version to wikipedia's version, but I can not do the converse. Can the converse hold? If not, which one is the right definition? Thanks.
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    Quote Originally Posted by zzzhhh View Post
    I got two definitions of \sigma-compact: one is from Munkres' "Topology" page 289, ex. 10, as the image below;


    the other is from wikipedia, which says \sigma-compact means the space is the union of countably many compact subspaces.
    I can deduce from Munkres' version to wikipedia's version, but I can not do the converse. Can the converse hold? If not, which one is the right definition? Thanks.
    The Wikipedia definition is what I have always seen as the standard definition of σ-compact. The Munkres definition looks superficially stronger, and it's not clear to me whether the two definitions are equivalent.
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