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

Authors : Lamia HARKAT, Abderrahmane BOUCHAİR, Smail KELAİAİA

Pages : 17-27

View : 36 | Download : 9

Publication Date : 2019-02-01

Article Type : Research Paper

Abstract :Let $X$ be a Tychonoff space, $Y$ an equiconnected space and $Cinsert ignore into journalissuearticles values(X,Y);$ 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 $Cinsert ignore into journalissuearticles values(X,Y);$ 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 $\pi_{Z}:Cinsert ignore into journalissuearticles values(X,Y);\rightarrow Cinsert ignore into journalissuearticles values(Z,Y);$ when both $Cinsert ignore into journalissuearticles values(X,Y);$ and $Cinsert ignore into journalissuearticles values(Z,Y);$ are endowed with the set open topology.
Keywords : Function space, set open topology, topology of uniform convergence on a family of sets, restriction map, Y compact