The Open-Point and Compact-Open Topology on C(X)

更新时间:2023-05-28

【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 X.The 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）| -={g∈C（X）:g-1（r）∩U≠φ},where f∈C（X）,C∈KC（X）={nonempty compact subsets of X},ε> 0,while U is an open subset of X and r∈R.The space C（X） equipped with the new topology Tkh which 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） X0 is Gδ-dense in X;（2） Ckh（X） is regular;（3） Ckh（X） is Tychonoff;（4） Ckh（X） is a topological group.We also show that if X is a Tychonoff space such that X0=Xc and Ckh（X） is regular space with countable pseudocharacter,then X is σ-compact.If X is a metrizable hemicompact countable space,then Ckh（X） is first countable.

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