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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 $\mathbb R^n$ 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.