- Hacettepe Journal of Mathematics and Statistics
- Vol: 48 Issue: 1
- Comparison of some set open and uniform topologies and some properties of the restriction maps
Comparison of some set open and uniform topologies and some properties of the restriction maps
Authors : Lamia Harkat, Abderrahmane Bouchair, Smail Kelaiaia
Pages : 17-27
View : 14 | Download : 6
Publication Date : 2019-02-01
Article Type : Research
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.Keywords : Function space, set open topology, topology of uniform convergence on a family of sets, restriction map, $Y$-compact