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H-closed-space

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absolutely-closed space

A Hausdorff space which, under any topological imbedding into an arbitrary Hausdorff space $Y$, is a closed set in $Y$. The characteristic property of an $H$-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 $H$-closed space is compact. If every closed subspace of a space is $H$-closed, then the space itself is compact. A theory has been developed for $H$-closed extensions of Hausdorff spaces.

References

[1] P.S. Aleksandrov, P.S. Urysohn, "Mémoire sur les espaces topologiques compacts" Verh. Akad. Wetensch. Amsterdam , 14 (1929)
[2] 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
[3] 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


Comments

References

[a1] J.R. Porter, R.G. Woods, "Extensions and absolutes of Hausdorff spaces" , Springer (1988)
How to Cite This Entry:
Absolutely closed space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Absolutely_closed_space&oldid=42457