A refinement on Michael’s characterization of paracompactness

Hung, H. H.

doi:10.1090/S0002-9939-1981-0620008-9

A refinement on Michael’s characterization of paracompactness
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by H. H. Hung PDF

Proc. Amer. Math. Soc. 83 (1981), 179-182 Request permission

Abstract:

It is proved that a ${T_1}$ space is paracompact (and Hausdorff) if (and only if) every well (but arbitrarily) ordered open cover has an open refinement in the form of a countable union of families, (each of which being well ordered by the naturally induced well order) the initial segments of each of which are cushioned in the corresponding initial segments of the cover, i.e., the closures of the unions of the initial segments of each of which are contained within the unions of the corresponding initial segments of the cover.

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