2025/08/20 by Soumajit Dey, Dey, Soumajit, Sudip Kumar Acharyya +3
Decision Sciences · #FOS: Mathematics #Fuzzy and Soft Set Theory #General Topology (math.GN) #Primary 54C40 #Secondary 46E30
paper · pdf · doi:10.48550/arxiv.2508.14651
openalex publication_date 2025/08/20 · openalex created_date 2025/10/16 · openalex updated_date 2026/08/03
If I is an ideal in the ring C(X) of all real valued continuous functions defined over a Tychonoff space X, then X is called I-pseudocompact if the set X∖ \bigcap Z[I] is a bounded subset of X. Corresponding to I, the mI-topology and uI-topology on C(X), generalizing the well-known m-topology and u-topology in C(X) respectively are already there in the literature. It is proved amongst others that the mI-topology is first countable if and only if the uI-topology= mI-topology on C(X) if and only if X is I-pseudocompact. A special case of this result on choosing I=C(X) reads: the u-topology and m-topology on C(X) coincide if and only if X is pseudocompact. It is established that the mI-topology on C(X) is second countable if and only if it is ℵ0-bounded if and only if X is compact, metrizable and I=C(X). Furthermore it is realized that the mI topology on C(X) is hemicompact if and only if it is σ-compact if and only if this topology is H-bounded if and only if X is finite and I=C(X).