arXiv subjects
Publications and source records attributed to E. A. Reznichenko.
It is proved that $P(βX)=βP(X)$ if and only if $P(X)$ is a pseudocompact space, where $P(X)$ is the space of Radon probability measures with the weak topology and $βX$ is the Stone--\v Cech compactification of $X$. A locally compact pseudocompact space $X$ is constructed such that $P(X)$ is not pseudocompact. Conditions are obtained under which $P(X)$ is pseudocompact.
A topological space is said to be cardinality homogeneous if every nonempty open subset has the same cardinality as the space itself. Let $X$ and $Y$ be cardinality homogeneous metric spaces of the same cardinality. If there exists a $δ$-surjective $d$-isometry between such equicardinal cardinality homogeneous metric spaces $X$ and $Y$, then there exists a bijective $(d+2δ)$-isometry between $X$ and $Y$. This result allows us to reduce the Dilworth--Tabor theorem to the Gevirtz--Omladič--Šemrl theorem on approximation by isometries and, in particular, to questions concerning the isometry of Banach spaces.
Generalizations of the theorems of Eberlein and Grothendieck on the precompactness of subsets of function spaces are considered: if $X$ is a countably compact space and $C_p(X)$ is a space of continuous functions in the pointwise topology convergence, then any countably compact subspace of the space $C_p(X)$ is precompact, that is, it has a compact closure. The paper provides an overview of the results on this topic. It is proved that if a pseudo-compact $X$ contains a dense Lindelof $Σ$-space, then pseudocompact subspaces of the space $C_p(X)$ are precompact. If $X$ is the product Cech complete spaces, then bounded subsets of the space $C_p(X)$ are precompact. Results on the continuity of separately continuous functions were also obtained.