0000000000928212
AUTHOR
Angelo Bella
A note on rank 2 diagonals
<p>We solve two questions regarding spaces with a (G<sub>δ</sub>)-diagonal of rank 2. One is a question of Basile, Bella and Ridderbos about weakly Lindelöf spaces with a G<sub>δ</sub>-diagonal of rank 2 and the other is a question of Arhangel’skii and Bella asking whether every space with a diagonal of rank 2 and cellularity continuum has cardinality at most continuum.</p>
A common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequality
AbstractWe present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\leqslant 2^{L(X)\unicode[STIX]{x1D712}(X)}$ (Arhangel’skiĭ) and $|X|\leqslant 2^{c(X)\unicode[STIX]{x1D712}(X)}$ (Hajnal–Juhász). This solves a question that goes back to Bell, Ginsburg and Woods’s 1978 paper (M. Bell, J.N. Ginsburg and R.G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79(1978), 37–45) and is mentioned in Hodel’s survey on Arhangel’skiĭ’s Theorem (R. Hodel, Arhangel’skii’s so…
Cardinal estimates involving the weak Lindelöf game
AbstractWe show that if X is a first-countable Urysohn space where player II has a winning strategy in the game $$G^{\omega _1}_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 ω 1 ( O , O D ) (the weak Lindelöf game of length $$\omega _1$$ ω 1 ) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a wi…
Infinite games and cardinal properties of topological spaces
Inspired by work of Scheepers and Tall, we use properties defined by topological games to provide bounds for the cardinality of topological spaces. We obtain a partial answer to an old question of Bell, Ginsburg and Woods regarding the cardinality of weakly Lindel¨of first-countable regular spaces and answer a question recently asked by Babinkostova, Pansera and Scheepers. In the second part of the paper we study a game-theoretic version of cellularity, a special case of which has been introduced by Aurichi. We obtain a game-theoretic proof of Shapirovskii’s bound for the number of regular open sets in an (almost) regular space and give a partial answer to a natural question about the produ…
Cardinal inequalities involving the Hausdorff pseudocharacter
We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter $H\psi(X)$. This invariant has the property $\psi_c(X)\leq H\psi(X)\leq\chi(X)$ for a Hausdorff space $X$. We show the cardinality of a Hausdorff space $X$ is bounded by $2^{pwL_c(X)H\psi(X)}$, where $pwL_c(X)\leq L(X)$ and $pwL_c(X)\leq c(X)$. This generalizes results of Bella and Spadaro, as well as Hodel. We show additionally that if $X$ is a Hausdorff linearly Lindel\"of space such that $H\psi(X)=\omega$, then $|X|\le 2^\omega$, under the assumption that either $2^{<\mathfrak{c}}=\mathfrak{c}$ or $\mathfrak{c}<\aleph_\omega$. The following game-theoretic result is shown: i…
Cardinal invariants of cellular Lindelof spaces
A space X is said to be cellular-Lindelof if for every cellular family $$\mathcal {U}$$ there is a Lindelof subspace L of X which meets every element of $$\mathcal {U}$$ . Cellular-Lindelof spaces generalize both Lindelof spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindelof monotonically normal space is Lindelof and that every cellular-Lindelof space with a regular $$G_\delta $$ -diagonal has cardinality at most $$2^\mathfrak {c}$$ . We also prove that every normal cellular-Lindelof first-countable space has cardinality at most continuum under $$2^{<\mathfrak {c}}=\mathfrak {c}$$ and that every normal cellular-Lindel…
Cardinal Invariants for the $G_\delta$ topology
We prove upper bounds for the spread, the Lindel\"of number and the weak Lindel\"of number of the $G_\delta$-topology on a topological space and apply a few of our bounds to give a short proof to a recent result of Juh\'asz and van Mill regarding the cardinality of a $\sigma$-countably tight homogeneous compactum.
Upper bounds for the tightness of the $$G_\delta $$-topology
We prove that if X is a regular space with no uncountable free sequences, then the tightness of its $$G_\delta $$ topology is at most the continuum and if X is, in addition, assumed to be Lindelof then its $$G_\delta $$ topology contains no free sequences of length larger then the continuum. We also show that, surprisingly, the higher cardinal generalization of our theorem does not hold, by constructing a regular space with no free sequences of length larger than $$\omega _1$$ , but whose $$G_\delta $$ topology can have arbitrarily large tightness.
Variations of selective separability II: Discrete sets and the influence of convergence and maximality
A space $X$ is called selectively separable(R-separable) if for every sequence of dense subspaces $(D_n : n\in\omega)$ one can pick finite (respectively, one-point) subsets $F_n\subset D_n$ such that $\bigcup_{n\in\omega}F_n$ is dense in $X$. These properties are much stronger than separability, but are equivalent to it in the presence of certain convergence properties. For example, we show that every Hausdorff separable radial space is R-separable and note that neither separable sequential nor separable Whyburn spaces have to be selectively separable. A space is called \emph{d-separable} if it has a dense $\sigma$-discrete subspace. We call a space $X$ D-separable if for every sequence of …
On the cardinality of almost discretely Lindelof spaces
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…