Approximation theorems for compactifications

Kotaro Mine

Colloquium Mathematicae (2011)

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

Abstract

top
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 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.

How to cite

top

Kotaro 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 -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.