Open Collections

Abstract

The set of all compactifications, K(X) of a locally compact, non-compact space X form a complete lattice with βX, the Stone-Čech compactification of X as its largest element, and αX, the one-point compactification of X as its smallest element. For any two locally compact, non-compact spaces X,Y, the lattices K(X), K(Y) are isomorphic if and only if βX - X and βY - Y are homeomorphic. βN is the Stone-Čech compactification of the countable infinite discrete space N. There is an isomorphism between the group of all homeomorphisms of βN and the group of all permutations of N; so βN has c homeomorphisms. The space N* =βN - N has 2c homeomorphisms. The cardinality of the set of orbits of the group of homeomorphisms of N* onto N* is 2c . If f is a homeomorphism of βN into itself, then Pk , the set of all k-periodic points of f is the closure of PkՈN in βN.