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A space X is countably compactly generated if it can be written as countable direct limit of compact Hausdorff spaces.

Are countably compactly generated spaces paracompact spaces? Do we have partition of unity for countably compactly generated spaces?

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  • $\begingroup$ Are you assuming that the compact Hausdorff spaces that enter the colimit are themselves separable, or even second countable? $\endgroup$
    – André Henriques
    Commented Dec 2, 2014 at 16:43
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    $\begingroup$ Asked already: math.stackexchange.com/q/1047888/94514 $\endgroup$
    – Ramiro de la Vega
    Commented Dec 2, 2014 at 16:45
  • $\begingroup$ If the final space is regular, the answer is yes: a countably compactly generated space is Lindelöf (or I am missing something ?), and a Lindelöf regular space is paracompact. $\endgroup$
    – Mathieu Baillif
    Commented Dec 3, 2014 at 8:47

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