Open Pseudocompactness in Topological Space

G Dharaniselvi, S Hariharan

Compact-Open Topology; Pseudocompactness; Submetrizability; Function Spaces; G-Delta Set; Sigma-Compact Space; Induced Mappings; Uniform Convergence.

Compactness is one of the most fundamental and influential concepts in general topology and functional analysis. Over the past century, various generalizations of compactness have been introduced in order to extend its applicability to broader classes of topological spaces. Among these, pseudocompactness and compact-open structures have played a significant role in the development of function space theory. The aim of this paper is to provide a comprehensive and detailed study of open compactness and open pseudocompactness in topological spaces, particularly in relation to function spaces of continuous real-valued functions. We analyze the structure of C(X), Ck(X), and Cps(X), examine their metrizability and submetrizability properties, study σ-compactness conditions, and investigate compactness-type equivalences. Special attention is given to induced mappings, Gδ-properties, separability, countability conditions, and the structural relationships between compact-open and pseudocompact-open topologies. The results presented here unify several known facts in function space topology and provide a broader structural perspective for further research in advanced topology and functional analysis.

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