Approximation theorems for compactifications

Abstract

We shall show several approximation theorems for the Hausdorff compactifications of metrizable spaces or locally compact Hausdorff spaces. It is shown that every compactification of the Euclidean n-space R n is the supremum of some compactifications homeomorphic to a subspace of R n+1 . Moreover, the following are equivalent for any connected locally compact Hausdorff space X:(i) X has no two-point compactifications, (ii) every compactification of X is the supremum of some compactifications whose remainder is homeomorphic to the unit closed interval or a singleton, (iii) every compactification of X is the supremum of some singular compactifications.We shall also give a necessary and sufficient condition for a compactification to be approximated by metrizable (or Smirnov) compactifications.2010 Mathematics Subject Classification: 54D35, 54D40, 46J10.