Peripherically-compact space

A topological space having a base of open sets with compact boundaries. A completely-regular peripherically-compact space $X$ has compactifications with zero-dimensional remainder (in the sense of the dimension ind, cf. Compactification; Remainder of a space; Dimension). If each compact set $A\subset X$ is contained in another compact set $B\subset X$ for which in $X$ there is a countable fundamental system of neighbourhoods (e.g., when $X$ is metrizable), then the peripheral compactness of $X$ is equivalent to the existence of compactifications of $X$ with zero-dimensional remainder.

References

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

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 [a1].

References

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