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 will 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 Pextensions 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.

A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y andOne-point P-extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the set of onepoint Tychonoff extensions of a (Tychonoff) space X (partially ordered by ) and the set of compact non-empty subsets of its outgrowth β X \ X (partially ordered by ⊆). This enables us to study the order-structure of various sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U (X) the set of all zero-sets of β X which miss X. Conjecture. For locally compact spaces X and Y the partially ordered sets (U (X), ⊆) and(U (Y ), ⊆) are order-isomorphic if and only if the spaces cl β X (β X \ υ X) and cl βY (βY \ υY ) are homeomorphic.

Let X be a completely regular topological space. We assign to each (set theoretic) ideal of X an (algebraic) ideal of C B (X), the normed algebra of continuous bounded complex valued mappings on X equipped with the supremum norm. We then prove several representation theorems for the assigned ideals of C B (X). This is done by associating a certain subspace of the Stone-Čech compactification βX of X to each ideal of X. This subspace of βX has a simple representation, and in the case when the assigned ideal of C B (X) is closed, coincides with its spectrum as a C * -subalgebra of C B (X). This in particular provides information about the spectrum of those closed ideals of C B (X) which have such representations. This includes the nonvanishing closed ideals of C B (X) whose spectrums are studied in great detail. Our representation theorems help to understand the structure of certain ideals of C B (X). This has been illustrated by means of various examples. Our approach throughout will be quite topological and makes use of the theory of the Stone-Čech compactification.

Abstract. We prove a commutative Gelfand-Naimark type theorem, by showing that the set Cs(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 C 0 (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 nonseparable, is moreover non-normal; in addition C 0 (Y ) = C 00 (Y ). When the underlying field of scalars is the complex numbers, the space Y coincides with the spectrum of the C * -algebra Cs(X). Further, we find the dimension of the algebra Cs(X).