arXiv subjects

Mikhail Al'perin

Publications and source records attributed to Mikhail Al'perin.

3 recordsLinked to original sources

arXiv2024

Dieudonné completeness of function spaces

A space is called Dieudonné complete if it is complete relative to the maximal uniform structure compatible with its topology. In this paper, we investigated when the function space $C(X,Y)$ of all continuous functions from a topological space $X$ into a uniform space $Y$ with the topology of uniform convergence on a family of subsets of $X$ is Dieudonné complete. Also we proved a generalization of the Eberlein-Šmulian theorem to the class of Banach spaces.

math.GN↗

arXiv2023

Generalization of the Grothendieck's theorem

In this paper, we have obtained a generalization of the Grothendieck's theorem for the space of continuous mappings $C_{λ,μ}(X,Y)$ where $Y$ is a complete uniform space with the uniformity $μ$ endowed with the topology of uniform convergence on the family $λ$ of subsets of $X$. A new topological game is defined - the Asanov-Velichko game, which makes it possible to single out a class of topological spaces of the Grothendieck type. The developed technique is used to generalize the Grothendieck theorem for the space of continuous mappings endowed with the set-open topology.

math.GN↗

arXiv2023

Embedding Theorems for function spaces

In this paper, we have proved results similar to Tychonoff's Theorem on embedding a space of functions with the topology of pointwise convergence into the Tychonoff product of topological spaces, but applied to the function space $C(X,Y)$ of all continuous functions from a topological space $X$ into a uniform space $Y$ with the topology of uniform convergence on a family of subsets of $X$ and with the (weak) set-open topology. We also investigated the following question: how the topological embedding of the space $C(X,Y)$ is related to algebraic structures (such as topological groups, topological rings and topological vector spaces) on $C(X,Y)$.

math.GN↗