# On the cardinality of almost discretely Lindelöf spaces

On the cardinality of almost discretely Lindelöf spaces A space is said to be almost discretely Lindelöf if every discrete subset can be covered by a Lindelöf subspace. Juhász et al. (Weakly linearly Lindelöf monotonically normal spaces are Lindelöf, preprint, arXiv:1610.04506 ) asked whether every almost discretely Lindelöf first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ 2 < c = c (which is a consequence of Martin’s Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juhász et al. (First-countable and almost discretely Lindelöf $$T_3$$ T 3 spaces have cardinality at most continuum, preprint, arXiv:1612.06651 ). We conclude with a few related results and questions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals

# On the cardinality of almost discretely Lindelöf spaces

, Volume 186 (2) – Oct 14, 2017
9 pages

/lp/springer_journal/on-the-cardinality-of-almost-discretely-lindel-f-spaces-1aqmDS1snX
Publisher
Springer Vienna
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-017-1112-4
Publisher site
See Article on Publisher Site

### Abstract

A space is said to be almost discretely Lindelöf if every discrete subset can be covered by a Lindelöf subspace. Juhász et al. (Weakly linearly Lindelöf monotonically normal spaces are Lindelöf, preprint, arXiv:1610.04506 ) asked whether every almost discretely Lindelöf first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ 2 < c = c (which is a consequence of Martin’s Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juhász et al. (First-countable and almost discretely Lindelöf $$T_3$$ T 3 spaces have cardinality at most continuum, preprint, arXiv:1612.06651 ). We conclude with a few related results and questions.

### Journal

Monatshefte f�r MathematikSpringer Journals

Published: Oct 14, 2017

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