The Open-Point and Compact-Open Topology on C(X)
The Open-Point and Compact-Open Topology on C（X）作者机构：College of Applied Science Beijing University of Technology
出 版 物：《Journal of Mathematical Research with Applications》 (数学研究及应用(英文))
年 卷 期：2020年第40卷第3期
页 面：305-312页
核心收录：
学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学]
基 金：Supported by the National Natural Science Foundation of China (Grant No. 11771029) the Natural Science Foundation of Beijing City (Grant No. 1202003)
主 题：C_p(X) C_k(X) Ckh(X) G_δ-dense
摘 要：In this note we define a new topology on C（X）,the set of all real-valued continuous functions on a Tychonoff space *** new topology on C（X） is the topology having subbase open sets of both kinds:[f,C,ε[={g E C（X）:|f（x）-g（x）| 0,while U is an open subset of X and r∈*** space C（X） equipped with the new topology Tkhwhich is stated above is denoted by Ckh（X）.Denote X0={x∈X:x is an isolated point of X} and Xc={x∈X:x has a compact neighborhood in X}.We show that if X is a Tychonoff space such that X0=Xc,then the following statements are equivalent:（1） X0is Gδ-dense in X;（2） Ckh（X） is regular;（3） Ckh（X） is Tychonoff;（4） Ckh（X） is a topological *** also show that if X is a Tychonoff space such that X0=Xcand Ckh（X） is regular space with countable pseudocharacter,then X is σ-*** X is a metrizable hemicompact countable space,then Ckh（X） is first countable.
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