The C-compact-open topology on function spaces

This paper studies the C-compact-open topology on the set C(X) of all realvalued continuous functions on a Tychonov space X and compares this topology with several well-known and lesser known topologies. We investigate the properties C-compact-open topology on the set C(X) such as submetrizable, metrizable, separable and second countability.

💡 Research Summary

The paper introduces and systematically studies the C‑compact‑open topology (denoted τ_C) on the space C(X) of all real‑valued continuous functions defined on a Tychonoff space X. A subset K⊆X is called C‑compact if for every f∈C(X) the image f(K) is compact in ℝ; equivalently, each continuous real‑valued function is bounded on K, so C‑compactness coincides with pseudocompactness of the subspace K. Using these sets as “test sets”, the authors define a subbasic neighbourhood of a function g by
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