A General Framework for Q−−P Spaces and Intermediate Hausdorff-Type Structures
Abstract
We introduce a unified framework of Q−−P spaces, designed to encompass a broad class of generalized topological structures arising from the interaction of two topological properties. We establish general permanence and transfer results, including heredity, productivity, additivity, and separation properties. Specializing to the Hausdorff setting, we investigate the intermediate classes C-Hausdorff, PS-Hausdorff, and CC-Hausdorff, and obtain the hierarchy T2 Longrightarrow Ps−Hausdorff Longrightarrow CC−Hausdorff Longrightarrow C−Hausdorff Longrightarrow T1. For C-Hausdorff spaces, we derive intrinsic characterizations and establish the equivalence with the simultaneous properties of k-Hausdorffness and KC. We further examine their connections with k-spaces, Fréchet spaces, local compactness, realcompactness, and ℵ0-boundedness. Finally, carefully constructed examples and counterexamples demonstrate the strictness of the hierarchy and the necessity of the hypothesis. Thus, the Q − P framework provides a unified perspective on generalized separation axioms and reveals a rich structure between classical Hausdorffness and T1-separation.
Related
- A General Framework for Operator Continuity between Riesz Spaces
- Refined proper descent for left completed sheaves on locally compact Hausdorff spaces
- Hausdorff Matrices on the BMOA and Bloch spaces
- Doubling, Poincaré and Gromov-Hausdorff precompactness for tamed spaces
- Cardinality Bounds for Hausdorff SDL Spaces
BibTeX
@misc{mitra2026general,
title = {{A General Framework for $Q\text--P$ Spaces and Intermediate Hausdorff-Type Structures}},
author = {Biswajit Mitra and Prodipta Ghosh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07945},
eprint = {2610.07945},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07945},
}
RIS
TY - UNPB
TI - A General Framework for Q−−P Spaces and Intermediate Hausdorff-Type Structures
AU - Biswajit Mitra
AU - Prodipta Ghosh
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.07945
UR - https://arxiv.org/abs/2610.07945
AB - We introduce a unified framework of Q−−P spaces, designed to encompass a broad class of generalized topological structures arising from the interaction of two topological properties. We establish general permanence and transfer results, including heredity, productivity, additivity, and separation properties. Specializing to the Hausdorff setting, we investigate the intermediate classes C-Hausdorff, PS-Hausdorff, and CC-Hausdorff, and obtain the hierarchy T_2 Longrightarrow Ps−Hausdorff Longrightarrow CC−Hausdorff Longrightarrow C−Hausdorff Longrightarrow T_1. For C-Hausdorff spaces, we derive intrinsic characterizations and establish the equivalence with the simultaneous properties of k-Hausdorffness and KC. We further examine their connections with k-spaces, Fréchet spaces, local compactness, realcompactness, and ℵ_0-boundedness. Finally, carefully constructed examples and counterexamples demonstrate the strictness of the hierarchy and the necessity of the hypothesis. Thus, the Q − P framework provides a unified perspective on generalized separation axioms and reveals a rich structure between classical Hausdorffness and T_1-separation.
ER -