TOPOLOGIES ON THE FUNCTION SPACE \(Y^X\) WITH VALUES IN A TOPOLOGICAL GROUP

Let \(Y^X\) denote the set of all functions from \(X\) to \(Y\). When \(Y\) is a topological space, various topologies can be defined on \(Y^X\). In this paper, we study these topologies within the framework of function spaces. To characterize different topologies and their properties, we employ gen...

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Published in:Ural mathematical journal Vol. 11; no. 1; pp. 77 - 93
Main Authors: Khatun, Kulchhum, Modak, Shyamapada
Format: Journal Article
Language:English
Published: Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics 29.07.2025
Subjects:
topological group, topological ideal, function space \(y^x\)
ISSN:2414-3952, 2414-3952
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Abstract Let \(Y^X\) denote the set of all functions from \(X\) to \(Y\). When \(Y\) is a topological space, various topologies can be defined on \(Y^X\). In this paper, we study these topologies within the framework of function spaces. To characterize different topologies and their properties, we employ generalized open sets in the topological space \(Y\). This approach also applies to the set of all continuous functions from \(X\) to \(Y\), denoted by \(C(X,Y)\), particularly when \(Y\) is a topological group. In investigating various topologies on both \(Y^X\) and \(C(X,Y)\), the concept of limit points plays a crucial role. The notion of a topological ideal provides a useful tool for defining limit points in such spaces. Thus, we utilize topological ideals to study the properties and consequences for function spaces and topological groups.
AbstractList Let \(Y^X\) denote the set of all functions from \(X\) to \(Y\). When \(Y\) is a topological space, various topologies can be defined on \(Y^X\). In this paper, we study these topologies within the framework of function spaces. To characterize different topologies and their properties, we employ generalized open sets in the topological space \(Y\). This approach also applies to the set of all continuous functions from \(X\) to \(Y\), denoted by \(C(X,Y)\), particularly when \(Y\) is a topological group. In investigating various topologies on both \(Y^X\) and \(C(X,Y)\), the concept of limit points plays a crucial role. The notion of a topological ideal provides a useful tool for defining limit points in such spaces. Thus, we utilize topological ideals to study the properties and consequences for function spaces and topological groups.
Author Modak, Shyamapada
Khatun, Kulchhum
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10.1016/j.topol.2009.06.022
10.2991/978-94-91216-35-0
10.31801/cfsuasmas.644689
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10.2140/pjm.1965.15.961
10.1016/S0166-8641(97)00257-5
10.4064/fm-91-1-5-10
10.1007/s40010-012-0039-3
10.2307/2312781
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SubjectTerms topological group, topological ideal, function space \(y^x\)
Title TOPOLOGIES ON THE FUNCTION SPACE \(Y^X\) WITH VALUES IN A TOPOLOGICAL GROUP
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