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Properties of locally H-closed spaces


Author: Mike Girou
Journal: Proc. Amer. Math. Soc. 113 (1991), 287-295
MSC: Primary 54D99
DOI: https://doi.org/10.1090/S0002-9939-1991-1047001-5
MathSciNet review: 1047001
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Abstract: This paper investigates the properties of locally $ H$-closed spaces with regard to extensions, subspaces, and functions. We solve the $ H$-closed extension remainder problem by showing that a space is locally $ H$-closed if and only if it has a $ \theta $-closed remainder in some $ H$-closed extension. In fact, an $ H$-closed space is Urysohn iff every $ H$-closed subspace is $ \theta $-closed. We solve the locally $ H$-closed subspace problem by giving a necessary and sufficient condition for a subspace of a locally $ H$-closed space to be locally $ H$-closed. In particular, an open subspace of a locally $ H$-closed space is locally $ H$-closed if and only if its boundary is a $ \theta $-closed subspace of its closure. An $ H$-closed space is shown to be compact if and only if every open subset is locally $ H$-closed. A retract of a locally $ H$-closed space is locally $ H$-closed.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1047001-5
Keywords: Closed space, locally $ H$-closed space, remainders, retracts
Article copyright: © Copyright 1991 American Mathematical Society

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