Skip to content
DraconiticResearch
Search Sources API
186,546 studies31 sourcesUpdated 53 min ago
Peer-reviewedCC BY 4.0Axioms

On SAC-P Spaces and Nowhere Densely Reflexive Properties

  • Dong-Fang Xie1
  • Wei-Xue Shi2
  • Wei-Feng Xuan3
  1. 1Department of Basic Courses, Suzhou City University, Suzhou 215104, China
  2. 2School of Mathematics, Nanjing University, Nanjing 210093, China
  3. 3School of Mathematics, Nanjing Audit University, Nanjing 211815, China
Open at sourceDOICite

Abstract

In this paper, we investigate how the properties of the nowhere dense subspaces reflect those of the entire space. We call a space X an SAC-P space if the set of accumulation points of any infinite family of mutually disjoint open subsets of X is a non-empty set having property P. We prove that for a regular space X, X is SAC-compact precisely when it is compact. In general, for a regular space X, X is SAC-P (here, P is a property which closed subsets inherit) exactly when X is feebly compact and each closed nowhere dense subspace of X has property P. Similar to discrete reflexivity, we introduce the notion of nowhere dense reflexivity. A property P is nowhere densely reflexive inA if each space X∈A has property P precisely when each closed nowhere dense subspace of X has property P. We prove that compactness, linear Lindelöfness and initial κ-compactness are all nowhere densely reflexive both in the class of feebly compact spaces and in that of dense-in-itself spaces. For topological groups, we prove that if G is a non-compact connected feathered topological group, then G is metrizable precisely when each closed nowhere dense subgroup of G is metrizable.

Related

  1. Reflexive and strictly convex Banach spaces without Property (H)
    Chunliu Feng, Geng Tian
    arXivPreprintCC BY-NC-SA 4.0Full textAnalysis & PDEsarXiv
  2. On densely defined linear continuous operators between function spaces
    Arkady Leiderman, Vesko Valov
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyarXiv
  3. Algebra properties for Besov spaces on unimodular Lie groups
    Joseph Feneuil
    Colloquium MathematicumPeer-reviewedCC BY 4.0Full text10 citedAnalysis & PDEsHAL
  4. On Certian Fine Properties of Sequence Spaces
    Hanuš Kameník
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  5. Lindelöf-type properties of spaces of Baire-one functions
    Alexander V. Osipov, Evgenii Reznichenko
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@article{xie2026sacp,
  title = {{On SAC-P Spaces and Nowhere Densely Reflexive Properties}},
  author = {Dong-Fang Xie and Wei-Xue Shi and Wei-Feng Xuan},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15100736},
  url = {https://www.mdpi.com/2075-1680/15/10/736},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - On SAC-P Spaces and Nowhere Densely Reflexive Properties
AU  - Dong-Fang Xie
AU  - Wei-Xue Shi
AU  - Wei-Feng Xuan
PY  - 2026
DA  - 2026/10/01
JO  - Axioms
DO  - 10.3390/axioms15100736
UR  - https://www.mdpi.com/2075-1680/15/10/736
AB  - In this paper, we investigate how the properties of the nowhere dense subspaces reflect those of the entire space. We call a space X an SAC-P space if the set of accumulation points of any infinite family of mutually disjoint open subsets of X is a non-empty set having property P. We prove that for a regular space X, X is SAC-compact precisely when it is compact. In general, for a regular space X, X is SAC-P (here, P is a property which closed subsets inherit) exactly when X is feebly compact and each closed nowhere dense subspace of X has property P. Similar to discrete reflexivity, we introduce the notion of nowhere dense reflexivity. A property P is nowhere densely reflexive inA if each space X∈A has property P precisely when each closed nowhere dense subspace of X has property P. We prove that compactness, linear Lindelöfness and initial κ-compactness are all nowhere densely reflexive both in the class of feebly compact spaces and in that of dense-in-itself spaces. For topological groups, we prove that if G is a non-compact connected feathered topological group, then G is metrizable precisely when each closed nowhere dense subgroup of G is metrizable.
SN  - 2075-1680
ER  -