# Article

Full entry | PDF   (0.5 MB)
Keywords:
Čech complete; H-closed; extension
Summary:
This paper investigates necessary and sufficient conditions for a space to have an H-closed extension with countable remainder. For countable spaces we are able to give two characterizations of those spaces admitting an H-closed extension with countable remainder. The general case is more difficult, however, we arrive at a necessary condition --- a generalization of Čech completeness, and several sufficient conditions for a space to have an H-closed extension with countable remainder. In particular, using the notation of Császár, we show that a space $X$ is a Čech $g$-space if and only if $X$ is $G_\delta$ in $\sigma X$ or equivalently if $EX$ is Čech complete. An example of a space which is a Čech $f$-space but not a Čech $g$-space is given answering a couple of questions of Császár. We show that if $X$ is a Čech $g$-space and $R(EX)$, the residue of $EX$, is Lindelöf, then $X$ has an H-closed extension with countable remainder. Finally, we investigate some natural generalizations of the residue to the class of all Hausdorff spaces.
References:
[1] Charalambous M.G.: Compactifications with countable remainder. Proc. Amer. Math. Soc. 78 (1980), no. 1, 127–131. DOI 10.1090/S0002-9939-1980-0548099-3 | MR 0548099 | Zbl 0439.54025
[2] Császár K.: Generalized Čech-complete spaces. Ann. Univ. Sci. Budapest Eötvös Sect. Math. 25 (1982), 229–238. MR 0683962
[3] Dickman R.F., Jr., Porter J.R., Rubin L.R.: Completely regular absolutes and projective objects. Pacific J. Math. 94 (1981), 277–295. DOI 10.2140/pjm.1981.94.277 | MR 0628580 | Zbl 0426.54005
[4] Engelking R.: General Topology. Polish Scientific Publishers, Warsaw, 1977. MR 0500780 | Zbl 0684.54001
[5] Frolík Z.: Generalization of the $G_{\delta }$-property of complete metric spaces. Czechoslovak Math. J. 10 (1960), 359–379. MR 0116305
[6] Gillman L., Jerison M.: Rings of Continuous Functions. Van Nostrand Reinhold, New York, 1960. MR 0116199 | Zbl 0327.46040
[7] Henriksen M.: Tychonoff spaces that have a compactification with a countable remainder. General Topology and its Relations to Modern Analysis and Algebra, IV (Proc. Fourth Prague Topological Sympos. Prague, 1976), Part B, Soc. Czech. Mathematicians and Physicists, Prague, 1977, pp. 164–167. MR 0454922
[8] Hoshina T.: Compactifications by adding a countable number of points. General Topology and its Relations to Modern Analysis and Algebra, IV Part B, Soc. Czech. Mathematicians and Physicists, Prague, 1977, pp. 168–169. MR 0461446 | Zbl 0387.54009
[9] Hoshina T.: Countable-points compactifications for metric spaces. Fund. Math. 103 (1979), 123–132. MR 0543874 | Zbl 0416.54018
[10] Hoshina T.: Countable-points compactifications of product spaces. Tsukuba J. Math. 6 (1982), 231–236. MR 0705116 | Zbl 0535.54009
[11] Morita K.: On bicompactifications of semibicompact spaces. Sci. R. Tokyo Bunrika Daigaku Sect. A 4 (1952), 200–207. MR 0052089 | Zbl 0049.39801
[12] Porter J.R.: Lattices of H-closed extensions. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 831–837. MR 0355964 | Zbl 0295.54028
[13] Porter J.R., Vermeer J.: Spaces with coarser Hausdorff topologies. Trans. Amer. Math. Soc. 289 (1985), no. 1, 59–71. DOI 10.1090/S0002-9947-1985-0779052-1 | MR 0779052
[14] Porter J.R., Votaw C.: H-closed extensions. II. Trans. Amer. Math. Soc. 202 (1975), 193–209. MR 0365493 | Zbl 0274.54012
[15] Porter J.R., Woods R.G.: Extensions and Absolutes of Hausdorff Spaces. Springer, Berllin, 1988. MR 0918341 | Zbl 0652.54016
[16] Terada T.: On countable discrete compactifications. General Topology Appl. 7 (1977), 321–327. MR 0500840 | Zbl 0364.54017
[17] Tikoo M.L.: Remainders of H-closed extensions. Topology Appl. 23 (1986), 117–128. DOI 10.1016/0166-8641(86)90033-7 | MR 0855451 | Zbl 0592.54024

Partner of