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Cardinality Bounds for Hausdorff SDL Spaces
Abstract
We establish the cardinal inequality |X|≤ 2t(X)Hψ(X) for every Hausdorff SDL space X, where t(X) and Hψ(X) denote the tightness and the Hausdorff pseudocharacter of X, respectively. Since both invariants are bounded by χ(X), this yields |X|≤ 2χ(X). As a consequence, every first-countable Hausdorff strongly cellular--Lindelöf space has cardinality at most the continuum. These results answer Questions~2.1 and~2.2 of Bella and Spadaro. An intermediate result is a uniform bounded-decomposition property for SDL spaces; in particular, their strict quasi--Lindelöf number satisfies sqL(X)≤ t(X).
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BibTeX
@misc{fernandes2026cardinality,
title = {{Cardinality Bounds for Hausdorff SDL Spaces}},
author = {Gabriel Fernandes and João Marcelo Maciel Messias},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00599},
eprint = {2609.00599},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00599},
}
RIS
TY - UNPB
TI - Cardinality Bounds for Hausdorff SDL Spaces
AU - Gabriel Fernandes
AU - João Marcelo Maciel Messias
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.00599
UR - https://arxiv.org/abs/2609.00599
AB - We establish the cardinal inequality |X|≤ 2^t(X)Hψ(X) for every Hausdorff SDL space X, where t(X) and Hψ(X) denote the tightness and the Hausdorff pseudocharacter of X, respectively. Since both invariants are bounded by χ(X), this yields |X|≤ 2^χ(X). As a consequence, every first-countable Hausdorff strongly cellular--Lindelöf space has cardinality at most the continuum. These results answer Questions~2.1 and~2.2 of Bella and Spadaro. An intermediate result is a uniform bounded-decomposition property for SDL spaces; in particular, their strict quasi--Lindelöf number satisfies sqL(X)≤ t(X).
ER -