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
References
- R. Arens and J. Dugundji, Topologies for function spaces. Pacific J. Math. 1, 5-31, 1951.
- A.V. Arhangel’skii, Topological Function Spaces, Kluwer Academic Publishers, 1992.
- A.V. Arhangel’skii, General topology III, Encyclopaedia of Mathematical Sciences 51, Springer-Verlag Berlin, 1995.
- A. Bouchair and S. Kelaiaia, Comparison of some set open topologies on $C(X,Y)$, Top. Appl. 178, 352-359, 2014.
- R. Engelking, General Topology, Polish Scientific Publishing, 1977.
- R. Fox, On fibre spaces II, Bull. Amer. Math. Soc. 49, 733-735, 1943.
- S. Kelaiaia, Propriétés de certaines topologies set-open sur $C(X)$, Thèse de Doctorat de l’Université de Rouen, 1995.
- R.A. McCoy and I. Ntantu, Countability properties of function spaces with set-open topology. Top. proc. 10, 329-345, 1985.
- R.A. McCoy and I. Ntantu, Topological properties of spaces of continuous functions. Lecture Note in Math. 1315, Springer Verlag Germany, 1988.
- S.E. Nokhrin and A.V. Osipov, On the coincidence of set-open and uniform topologies. Proc. steklov Inst. Math. Suppl. 3, 184-191, 2009. Top. Appl. 150, 255-265, 2005.
- A.V. Osipov, Uniformity of uniform convergence on the family of sets, Top. Proc. 50, 79-86, 2017.
- S. Önal and C. Vural, Some cardinal invariants on the space $C_{\alpha}(X,Y)$, Top. Appl. 150, 255-265, 2005.
Abstract
References
- R. Arens and J. Dugundji, Topologies for function spaces. Pacific J. Math. 1, 5-31, 1951.
- A.V. Arhangel’skii, Topological Function Spaces, Kluwer Academic Publishers, 1992.
- A.V. Arhangel’skii, General topology III, Encyclopaedia of Mathematical Sciences 51, Springer-Verlag Berlin, 1995.
- A. Bouchair and S. Kelaiaia, Comparison of some set open topologies on $C(X,Y)$, Top. Appl. 178, 352-359, 2014.
- R. Engelking, General Topology, Polish Scientific Publishing, 1977.
- R. Fox, On fibre spaces II, Bull. Amer. Math. Soc. 49, 733-735, 1943.
- S. Kelaiaia, Propriétés de certaines topologies set-open sur $C(X)$, Thèse de Doctorat de l’Université de Rouen, 1995.
- R.A. McCoy and I. Ntantu, Countability properties of function spaces with set-open topology. Top. proc. 10, 329-345, 1985.
- R.A. McCoy and I. Ntantu, Topological properties of spaces of continuous functions. Lecture Note in Math. 1315, Springer Verlag Germany, 1988.
- S.E. Nokhrin and A.V. Osipov, On the coincidence of set-open and uniform topologies. Proc. steklov Inst. Math. Suppl. 3, 184-191, 2009. Top. Appl. 150, 255-265, 2005.
- A.V. Osipov, Uniformity of uniform convergence on the family of sets, Top. Proc. 50, 79-86, 2017.
- S. Önal and C. Vural, Some cardinal invariants on the space $C_{\alpha}(X,Y)$, Top. Appl. 150, 255-265, 2005.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Mathematics |
| Authors | |
| Publication Date | February 1, 2019 |
| Published in Issue | Year 2019 Volume: 48 Issue: 1 |