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.
📌 3 Maddede Bu Makale
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Amaç & Hipotez: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 $X$.
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Temel Bulgular & Veri: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 $X$.
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Sonuç & Katkı: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.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Function space · set open topology · topology of uniform convergence on a family of sets
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.
Harkat, L., Bouchair, A., Kelaiaia, S. (2019). Comparison of some set open and uniform topologies and some properties of the restriction maps. Hacettepe Journal of Mathematics and Statistics, Volume: 48 Issue: 1. https://ajindex.com/articles/1402829
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.