Remainder of a space

The set , where is a compactification of . The properties of a remainder depend strongly on those of : A remainder is compact if and only if is locally compact; the existence of a zero-dimensional remainder depends broadly on whether has the peripheral compactness property; if there exists a metrizable compactification of with remainder of dimension , then has an open base in which the intersection of the boundaries of any disjoint sets is compact; etc. If every connected compact subset of consists of a single point (for example, if ), the remainder is said to be punctiform. If there exists at least one compactification with punctiform remainder, then one of these compactifications, , is maximal, and, moreover, it is the minimal perfect extension of .

Comments

Some important properties, such as compactness, local compactness, paracompactness, the Lindelöf property, are possessed by all remainders of or by none. When the remainders of have such a property, is said to have the property "at infinity" .

References

[a1]	H.N. Inasaridze, "A generalization of perfect mappings" Soviet Math. Dokl. , 7 : 3 (1966) pp. 620–622 Dokl. Akad. Nauk SSSR , 168 (1966) pp. 266–268