# 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)