Embedding Theorems for function spaces

TL;DR

This paper extends classical embedding theorems to function spaces with various topologies, exploring their topological and algebraic properties and relationships.

Contribution

It proves new embedding theorems for function spaces with different topologies and examines their connection to algebraic structures.

Findings

Embedding theorems for $C(X,Y)$ with pointwise and set-open topologies.

Relationship between topological embeddings and algebraic structures.

Analysis of how embeddings influence algebraic properties of function spaces.

Abstract

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 $C(X,Y)$.