Approximation theorems for compactifications

Kotaro Mine¹

¹ Institute of Mathematics University of Tsukuba Tsukuba, 305-8571, Japan

Colloquium Mathematicum, Tome 122 (2011) no. 1, pp. 93-101.

Résumé

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 $\mathbb R^n$ is the supremum of some compactifications homeomorphic to a subspace of $\mathbb R^{n+1}$. 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.

Détail

DOI : 10.4064/cm122-1-9

Keywords: shall several approximation theorems hausdorff compactifications metrizable spaces locally compact hausdorff spaces shown every compactification euclidean n space mathbb supremum compactifications homeomorphic subspace mathbb moreover following equivalent connected locally compact hausdorff space nbsp has two point compactifications every compactification supremum compactifications whose remainder homeomorphic unit closed interval singleton iii every compactification supremum singular compactifications shall necessary sufficient condition compactification approximated metrizable smirnov compactifications