# Approximation theorems for compactifications

Colloquium Mathematicae (2011)

- Volume: 122, Issue: 1, page 93-101
- ISSN: 0010-1354

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topKotaro Mine. "Approximation theorems for compactifications." Colloquium Mathematicae 122.1 (2011): 93-101. <http://eudml.org/doc/286261>.

@article{KotaroMine2011,

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 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.},

author = {Kotaro Mine},

journal = {Colloquium Mathematicae},

keywords = {Compactification; Smirnov compactification; singular compactification; remainder; supremum of compactifications; unital Banach algebra},

language = {eng},

number = {1},

pages = {93-101},

title = {Approximation theorems for compactifications},

url = {http://eudml.org/doc/286261},

volume = {122},

year = {2011},

}

TY - JOUR

AU - Kotaro Mine

TI - Approximation theorems for compactifications

JO - Colloquium Mathematicae

PY - 2011

VL - 122

IS - 1

SP - 93

EP - 101

AB - 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.

LA - eng

KW - Compactification; Smirnov compactification; singular compactification; remainder; supremum of compactifications; unital Banach algebra

UR - http://eudml.org/doc/286261

ER -

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