View source for H-closed-space

''absolutely-closed space''

A [[Hausdorff space|Hausdorff space]] which, under any topological imbedding into an arbitrary Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046000/h0460002.png" />, is a closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046000/h0460003.png" />. The characteristic property of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046000/h0460004.png" />-closed space is that any open covering of the space contains a finite subfamily the closures of the elements of which cover the space. A regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046000/h0460005.png" />-closed space is compact. If every closed subspace of a space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046000/h0460006.png" />-closed, then the space itself is compact. A theory has been developed for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046000/h0460007.png" />-closed extensions of Hausdorff spaces.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   P.S. Urysohn,   "Mémoire sur les espaces topologiques compacts"  ''Verh. Akad. Wetensch. Amsterdam'' , '''14'''  (1929)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.D. Iliadis,   S.V. Fomin,   "The method of centred systems in the theory of topological spaces"  ''Russ. Math. Surveys'' , '''21''' :  4  (1966)  pp. 37–62  ''Uspekhi Mat. Nauk'' , '''21''' :  4  (1966)  pp. 47–76</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Malykhin,   V.I. Ponomarev,   "General topology (set-theoretic trend)"  ''J. Soviet Math.'' , '''7'''  (1977)  pp. 587–653  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''13'''  (1975)  pp. 149–230</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.R. Porter,   R.G. Woods,   "Extensions and absolutes of Hausdorff spaces" , Springer  (1988)</TD></TR></table>