View source for Peripherically-compact space

A [[Topological space|topological space]] having a [[Base|base]] of open sets with compact boundaries. A completely-regular peripherically-compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072240/p0722401.png" /> has compactifications with zero-dimensional remainder (in the sense of the dimension ind, cf. [[Compactification|Compactification]]; [[Remainder of a space|Remainder of a space]]; [[Dimension|Dimension]]). If each compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072240/p0722402.png" /> is contained in another compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072240/p0722403.png" /> for which in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072240/p0722404.png" /> there is a countable fundamental system of neighbourhoods (e.g., when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072240/p0722405.png" /> is metrizable), then the peripheral compactness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072240/p0722406.png" /> is equivalent to the existence of compactifications of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072240/p0722407.png" /> with zero-dimensional remainder.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Freudenthal,   "Neuaufbau der Endentheorie"  ''Ann. of Math.'' , '''43'''  (1942)  pp. 261–279</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Freudenthal,   "Kompaktisierungen und Bikompaktisierungen"  ''Indag. Math.'' , '''13''' :  2  (1951)  pp. 184–192</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.G. Sklyarenko,   "Bicompact extensions of semibicompact spaces"  ''Dokl. Akad. Nauk. SSSR'' , '''120''' :  6  (1958)  pp. 1200–1203  (In Russian)</TD></TR></table>



====Comments====
These spaces are also called rim-compact spaces.

Spaces with the property that every compact subset is contained in a compact subset with a countable neighbourhood base are called spaces of countable type, see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.R. Isbell,   "Uniform spaces" , Amer. Math. Soc.  (1964)  pp. Chapt. 7</TD></TR></table>