EN
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 ℝⁿ 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.