We discuss some notions of compactness relative to a specified family F of subsets of some topological space X . In particular, we relativize to F the notions ofD-compactness, CAPλ, and [μ, λ]-compactness. The two most interesting particular cases of our construction appear to be the following ones. (1) The case in which F is the family of all singletons of X , in which case we get back the mor...

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F ⊆ X and every point x ∈ X \ F , there exist f ∈ Cp(X,G) and g ∈ G \ {e} such that f(x) = g and f(F ) ⊆ {e}; (b) G-regular provided that t...

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–Čech compactification βX (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing seq...

We present an example of a separable metrizable topological group G having the property that no remainder of it is (topologically) homogeneous. 1. Introduction. All topological spaces under discussion are Tychonoff. A space X is homogeneous if for any two points x, y ∈ X there is a homeomorphism h from X onto itself such that h(x) = y. If bX is a com-pactification of a space X, then bX \ X is c...

The main result of this paper is a generalization of a theorem of R. S. Phillips [6] on the nonexistence of projections of l°° onto c0. If S is a locally compact Hausdorff space then let C(S) denote the space of bounded continuous real (or complex) valued functions on 5; also, let Co(S) be those functions in C(S) which vanish at infinity. If N is the space of positive integers with the discrete...

The notion of a tensor product of topological groups and modules is important in theory of topological groups, algebraic number theory. The tensor product of compact zero-dimensional modules over a pseudocompact algebra was introduced in [B] and for the commutative case in [GD], [L]. The notion of a tensor product of abelian groups was introduced in [H]. The tensor product of modules over commu...

We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product [X × H × Y ]σ of a topological group H over topological spaces X, Y with a continuous sandwich function σ : Y × X → H. We prove that a simple topological semigroup S is a topological paragroup if one of the following conditions is satisfied: (1) S is completely simple and the maximal subgrou...

We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p, q). We prove that each topological semigroup S with pseudocompact square contains no dense copy of C(p, q). On the other hand, we construct a consistent example of a Tychonov countably compact semigroup containing a copy of C(p, q). In this paper we study the structural prope...

We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product [X × H × Y ]σ of a topological group H over topological spaces X, Y with a continuous sandwich function σ : Y × X → H. We prove that a simple topological semigroup S is a topological paragroup if one of the following conditions is satisfied: (1) S is completely simple and the maximal subgrou...