Comparison of some set open and uniform topologies and some properties of the restriction maps

Abstract

Let $X$ be a Tychonoff space, $Y$ an equiconnected space and $C(X,Y)$ be the set of all continuous functions from $X$ to $Y$. In this paper, we provide a criterion for the coincidence of set open and uniform topologies on $C(X,Y)$ when these topologies are defined by a family $\alpha$ consisting of $Y$-compact subsets of $X$. For a subspace $Z$ of a topological space $X$, we also study the continuity and the openness of the restriction map $\pi_{Z}:C(X,Y)\rightarrow C(Z,Y)$ when both $C(X,Y)$ and $C(Z,Y)$ are endowed with the set open topology.