# Compactification-like extensions

- 2011

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topM. R. Koushesh. Compactification-like extensions. 2011. <http://eudml.org/doc/285950>.

@book{M2011,

abstract = {Let X be a space. A space Y is called an extension of X if Y contains X as a dense subspace. For an extension Y of X the subspace Y∖X of Y is called the remainder of Y. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X pointwise. For two (equivalence classes of) extensions Y and Y' of X let Y ≤ Y' if there is a continuous mapping of Y' into Y which fixes X pointwise. Let 𝓟 be a topological property. An extension Y of X is called a 𝓟-extension of X if it has 𝓟. If 𝓟 is compactness then 𝓟-extensions are called compactifications. The aim of this article is to introduce and study classes of extensions (which we call compactification-like 𝓟-extensions, where 𝓟 is a topological property subject to some mild requirements) which resemble the classes of compactifications of locally compact spaces. We formally define compactification-like 𝓟-extensions and derive some of their basic properties, and in the case when the remainders are countable, we characterize spaces having such extensions. We then consider the classes of compactification-like 𝓟-extensions as partially ordered sets. This consideration leads to some interesting results which characterize compactification-like 𝓟-extensions of a space among all its Tychonoff 𝓟-extensions with compact remainder. Furthermore, we study the relations between the order-structure of classes of compactification-like 𝓟-extensions of a Tychonoff space X and the topology of a certain subspace of its outgrowth βX∖X. We conclude with some applications, including an answer to an old question of S. Mrówka and J. H. Tsai: For what pairs of topological properties 𝓟 and 𝓠 is it true that every locally-𝓟 space with 𝓠 has a one-point extension with both 𝓟 and 𝓠? An open question is raised.},

author = {M. R. Koushesh},

keywords = {Stone-Čech compactification; compactification-like extension; minimal extension; optimal extension; tight extension; -point extension; -point compactification; countable-point extension; countable-point compactification; countable extension; countable compactification; Mrówka’s condition (W); compactness-like topological property},

language = {eng},

title = {Compactification-like extensions},

url = {http://eudml.org/doc/285950},

year = {2011},

}

TY - BOOK

AU - M. R. Koushesh

TI - Compactification-like extensions

PY - 2011

AB - Let X be a space. A space Y is called an extension of X if Y contains X as a dense subspace. For an extension Y of X the subspace Y∖X of Y is called the remainder of Y. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X pointwise. For two (equivalence classes of) extensions Y and Y' of X let Y ≤ Y' if there is a continuous mapping of Y' into Y which fixes X pointwise. Let 𝓟 be a topological property. An extension Y of X is called a 𝓟-extension of X if it has 𝓟. If 𝓟 is compactness then 𝓟-extensions are called compactifications. The aim of this article is to introduce and study classes of extensions (which we call compactification-like 𝓟-extensions, where 𝓟 is a topological property subject to some mild requirements) which resemble the classes of compactifications of locally compact spaces. We formally define compactification-like 𝓟-extensions and derive some of their basic properties, and in the case when the remainders are countable, we characterize spaces having such extensions. We then consider the classes of compactification-like 𝓟-extensions as partially ordered sets. This consideration leads to some interesting results which characterize compactification-like 𝓟-extensions of a space among all its Tychonoff 𝓟-extensions with compact remainder. Furthermore, we study the relations between the order-structure of classes of compactification-like 𝓟-extensions of a Tychonoff space X and the topology of a certain subspace of its outgrowth βX∖X. We conclude with some applications, including an answer to an old question of S. Mrówka and J. H. Tsai: For what pairs of topological properties 𝓟 and 𝓠 is it true that every locally-𝓟 space with 𝓠 has a one-point extension with both 𝓟 and 𝓠? An open question is raised.

LA - eng

KW - Stone-Čech compactification; compactification-like extension; minimal extension; optimal extension; tight extension; -point extension; -point compactification; countable-point extension; countable-point compactification; countable extension; countable compactification; Mrówka’s condition (W); compactness-like topological property

UR - http://eudml.org/doc/285950

ER -

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