Approximation theorems for compactifications

Volume 122 / 2011

Kotaro Mine Colloquium Mathematicum 122 (2011), 93-101 MSC: 54D35, 54D40, 46J10. DOI: 10.4064/cm122-1-9


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.


  • Kotaro MineInstitute of Mathematics
    University of Tsukuba
    Tsukuba, 305-8571, Japan

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