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Compactification-like extensions

100%

Koushesh M.

Instytut Matematyczny Polskiej Akademii Nauk

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.

The Banach algebra of continuous bounded functions with separable support

96%

Koushesh M.

Studia Mathematica

2012

tom 210

nr 3

227-237

We prove a commutative Gelfand-Naimark type theorem, by showing that the set $C_{s}(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space X (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to C₀(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y, which we explicitly construct as a subspace of the Stone-Čech compactification of X, is countably compact, and if X is non-separable, is moreover non-normal; in addition C₀(Y) = C₀₀(Y). When the underlying field of scalars is the complex numbers, the space Y coincides with the spectrum of the C*-algebra $C_{s}(X)$. Further, we find the dimension of the algebra $C_{s}(X)$.

Ograniczanie wyników

1 Studia Mathematica

2 Koushesh M.

1 2012

1 2011

Wyniki wyszukiwania

Compactification-like extensions

The Banach algebra of continuous bounded functions with separable support