When is a space Menger at infinity?
We try to characterize those Tychonoff spaces $X$ such that $βX \setminus X$ has the Menger property.
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arXiv subjects
Angelo Bella
Publications and source records attributed to Angelo Bella.
On a game theoretic cardinality bound
The main purpose of the paper is the proof of a cardinal inequality for a space with points $G_δ$, obtained with the help of a long version of the Menger game. This result improves a similar one of Scheepers and Tall.
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Topological games and productively countably tight spaces
The two main results of this work are the following: if a space $X$ is such that player II has a winning strategy in the game $\gone(Ω_x, Ω_x)$ for every $x \in X$, then $X$ is productively countably tight. On the other hand, if a space is productively countably tight, then $\sone(Ω_x, Ω_x)$ holds for every $x \in X$. With these results, several other results follow, using some characterizations made by Uspenskii and Scheepers.
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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 Lindelöf 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 productivity of a game strengthening of the countable chain condition that was introduced by Aurichi. As a final application of our results we prove that the Hajnal-Juhász bound for the cardinality of a first-countable ccc Hausdorff space is true for almost regular (non-Hausdorff) spaces.
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Some observations on compact indestructible spaces
Inspired by a recent work of Dias and Tall, we show that a compact indestructible space is sequentially compact. We also prove that a Lindelof Hausdorff indestructible space has the finite derived set property and a compact Hausdorff indestructible space is pseudoradial.
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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 $(D_n : n\inω)$ one can pick finite (respectively, one-point) subsets $F_n\subset D_n$ such that $\bigcup_{n\inω}F_n$ 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 $σ$-discrete subspace. We call a space $X$ D-separable if for every sequence of dense subspaces $(D_n : n\inω)$ one can pick discrete subsets $F_n\subset D_n$ such that $\bigcup_{n\inω}F_n$ is dense in $X$. Although $d$-separable spaces are often also $D$-separable (this is the case, for example, with linearly ordered $d$-separable or stratifiable spaces), we offer three examples of countable non-$D$-separable spaces. It is known that d-separability is preserved by arbitrary products, and that for every $X$, the power $X^{d(X)}$ is d-separable. We show that D-separability is not preserved even by finite products, and that for every infinite $X$, the power $X^{2^{d(X)}}$ is not D-separable. However, for every $X$ there is a $Y$ such that $X\times Y$ is D-separable. Finally, we discuss selective and D-separability in the presence of maximality. For example, we show that (assuming ${\mathfrak d}=\mathfrak c$) there exists a maximal regular countable selectively separable space, and that (in ZFC) every maximal countable space is D-separable (while some of those are not selectively separable). However, no maximal space satisfies the natural game-theoretic strengthening of D-separability.
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P-spaces and the Whyburn property
We investigate the Whyburn and weakly Whyburn property in the class of $P$-spaces, that is spaces where every countable intersection of open sets is open. We construct examples of non-weakly Whyburn $P$-spaces of size continuum, thus giving a negative answer under CH to a question of Pelant, Tkachenko, Tkachuk and Wilson. In addition, we show that the weak Kurepa Hypothesis (a set-theoretic assumption weaker than CH) implies the existence of a non-weakly Whyburn $P$-space of size $\aleph_2$. Finally, we consider the behavior of the above-mentioned properties under products; we show in particular that the product of a Lindelöf weakly Whyburn P-space and a Lindelöf Whyburn $P$-space is weakly Whyburn, and we give a consistent example of a non-Whyburn product of two Lindelöf Whyburn $P$-spaces.
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