Search results for " Lindelof space"
showing 2 items of 2 documents
On closures of discrete sets
2018
The depth of a topological space $X$ ($g(X)$) is defined as the supremum of the cardinalities of closures of discrete subsets of $X$. Solving a problem of Mart\'inez-Ruiz, Ram\'irez-P\'aramo and Romero-Morales, we prove that the cardinal inequality $|X| \leq g(X)^{L(X) \cdot F(X)}$ holds for every Hausdorff space $X$, where $L(X)$ is the Lindel\"of number of $X$ and $F(X)$ is the supremum of the cardinalities of the free sequences in $X$.
On the cardinality of almost discretely Lindelof spaces
2016
A space is said to be almost discretely Lindelof if every discrete subset can be covered by a Lindelof subspace. Juhasz et al. (Weakly linearly Lindelof monotonically normal spaces are Lindelof, preprint, arXiv:1610.04506 ) asked whether every almost discretely Lindelof first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ (which is a consequence of Martin’s Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juhasz et al. (First-countable and almost discretely Lindelof $$T_3$$ spaces have cardinality at most continuum, preprint, arXiv:1612.06651 ). We conclude with a few rel…