A.V. Arhangel'skii

Weak developments and metrization

Abstract The notions of a weak k -development and of a weak development, defined in terms of sequences of open covers, were recently introduced by the first and the third authors. The first notion was applied to extend in an interesting way Michaels Theorem on double set-valued selections. The second notion is situated between that of a development and of a base of countable order. To see that a space with a weak development has a base of countable order, we use the classical works of H.H. Wicke and J.M. Worrell. We also introduce and study the new notion of a sharp base, which is strictly weaker than that of a uniform base and strictly stronger than that of a base of countable order and of a weakly uniform base, and which is strongly connected to the notion of a weak development. Several examples are exhibited to prove that the new notions do not coincide with the old ones. In short, our results show that the notions of a weak development and of a sharp base fit very well into already existing system of generalized metrizability properties defined in terms of sequences of open covers or bases. Several open questions are formulated.

Sharp bases and weakly uniform bases versus point-countable bases

Abstract A base B for a topological space X is said to be sharp if for every x∈X and every sequence (U n ) n∈ω of pairwise distinct elements of B with x∈U n for all n the set {⋂ i U i : n∈ω} forms a base at x . Sharp bases of T 0 -spaces are weakly uniform. We investigate which spaces with sharp bases or weakly uniform bases have point-countable bases or are metrizable. In particular, Davis, Reed, and Wage had constructed in a 1976 paper a consistent example of a Moore space with weakly uniform base, but without a point-countable base. They asked whether such an example can be constructed in ZFC. We partly answer this question by showing that under CH, every first-countable space with a weakly uniform base and at most ℵ ω isolated points has a point-countable base.

Topological groups and C-embeddings

Abstract The notion of a Moscow space is applied to the study of some problems of topological algebra, following an approach introduced by A.V. Arhangelskii [Comment. Math. Univ. Carolin. 41 (2000) 585–595]. In particular, many new, and, it seems, unexpected, solutions to the equation νX×νY=ν(X×Y) are identified. We also find new large classes of topological groups G , for which the operations in G can be extended to the Dieudonne completion of the space G in such a way that G becomes a topological subgroup of the topological group μG . On the other hand, it was shown by A.V. Arhangelskii [Comment. Math. Univ. Carolin. 41 (2000) 585–595] that there exists an Abelian topological group G for which such an extension is impossible (this provided an answer to a question of V.G. Pestov and M.G. Tkacenko, dating back to 1985). Some new open questions are formulated.

Projective σ-compactness, ω1-caliber, and Cp-spaces

Abstract A space is called projectively σ -compact, if every separable metrizable continuous image of this space is σ -compact. In particular, we establish when C p ( X ) is projectively σ -compact. In the last section, a theorem on cardinality of Lindelof spaces is proved and then applied to obtain a result in C p -theory.

On a theorem of Grothendieck in Cp-theory

Abstract A theorem of Grothendieck is extended in several directions. In particular, it is proved that if X is a Lindelof ∑-space, then the closure of every relatively countably compact subset of C p ( X ) is compact. We also investigate when the closure of every pseudocompact subspace of C p ( X ) is compact, and when every pseudocompact subset of C p ( X ) is itself compact, or even belongs to a given class of compacta. New open questions are formulated.

Homogeneity of powers of spaces and the character

A space is said to be power-homogeneous if some power of it is homogeneous. We prove that if a Hausdorff space X of point-countable type is power-homogeneous, then, for every infinite cardinal r, the set of points at which X has a base of cardinality not greater than T, is closed in X. Every power-homogeneous linearly ordered topological space also has this property. Further, if a linearly ordered space X of point-countable type is power-homogeneous, then X is first countable.

Normality and dense subspaces

In the first section of this paper, using certain powerful results in Cp-theory, we show that there exists a nice linear topological space X of weight ω 1 such that no dense subspace of X is normal. In the second and third sections a natural generalization of normality, called dense normality, is considered. In particular, it is shown in section 2 that the space R c is not normal on some countable dense subspace of it, while it is normal on some other dense subspace. An example of a Tychonoff space X, which is not densely normal on a dense separable metrizable subspace, is constructed. In section 3, a link between dense normality and relative countable compactness is established. In section 4 the result of section 1 is extended to densely normal spaces.

Embeddings in Cp-spaces

Abstract We survey C p -theory from the point of view provided by the following question: when can a topological space be embedded into C p ( X ), where X belongs to a given class of spaces? The question turns out to have interesting links to many general and concrete situations. In particular, we discuss from this position t -, and l -equivalences, continuous extenders, a theorem of Grothendieck, and the topological structure of compact subspaces of C p ( X ), where X is Lindelof. Many open problems, arising naturally under this approach, are presented, some of them new. Some results in the paper are also formulated for the first time, in particular, Theorems 2.29, 2.30, 2.38, 3.24, 5.3, 5.24, 5.25 and Corollaries 2.28, 3.23.

Strongly τ-pseudocompact spaces

Abstract A new notion of a strongly τ-pseudocompact space is introduced, and its relation to the notion of initially τ-compact space is studied. A theorem of J.F. Kennison (Kennison, 1962) is generalized to this case, which leads to new interesting questions. In particular, it is proved that every strongly 2 ω -pseudocompact space of countable tightness is compact, and that every strongly 2 ω -pseudocompact space is initially τ + -compact. It is also established that every ω 1 -pseudocompact topological group of countable tightness is metrizable and, therefore, compact. Several open problems are formulated.