Countably compact weakly Whyburn spaces

The weak Whyburn property is a generalization of the classical sequential property that has been studied by many authors. A space X is weakly Whyburn if for every non-closed set A⊂X there is a subset B⊂A such that B⎯⎯⎯⎯∖A is a singleton. We prove that every countably compact Urysohn space of cardinality smaller than the continuum is weakly Whyburn and show that, consistently, the Urysohn assumption is essential. We also give conditions for a (countably compact) weak Whyburn space to be pseudoradial and construct a countably compact weakly Whyburn non-pseudoradial regular space, which solves a question asked by Bella in private communication

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 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-Juh´asz bound for the cardinalityof a first-countable ccc Hausdorff space is true for almost regular (non-Hausdorff) space

Comparing weak versions of separability

Our aim is to investigate spaces with sigma-discrete and meager dense sets, as well as selective versions of these properties. We construct numerous examples to point out the differences between these classes while answering questions of Tkachuk [30], Hutchinson [17] and the authors of [8]

Selective versions of chain condition type properties

We study selective and game-theoretic versions of properties like the ccc, weak Lindelöfness and separability, giving various characterizations of them and exploring connections between these properties and some classical cardinal invariants of the continuum

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 is an element of omega) one can pick finite (respectively, one-point) subsets F(n) subset of D(n), such that boolean OR(n is an element of 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 d-separable if it has a dense sigma-discrete subspace. We call a space X D-separable if for every sequence of dense subspaces (D(n): n is an element of omega) one can pick discrete subsets F(n) subset of D(n) such that boolean OR(n is an element of omega) 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(2d(X)) is not D-separable. However, for every X there is a Y such that X x Y is D-separable. Finally, we discuss selective and D-separability in the presence of maximality. For example, we show that (assuming d = 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

A new class of spaces with all finite powers Lindelof

We consider a new class of open covers and classes of spaces defined from them, called "iota spaces". We explore their relationship with epsilon-spaces (that is, spaces having all finite powers Lindelof) and countable network weight. An example of a hereditarily epsilon-space whose square is not hereditarily Lindelof is provided answering a question from

Infinite games and chain conditions

We apply the theory of infinite two-person games to two well-known problems in topology: Suslin's Problem and Arhangel'skii's problem on Gδ covers of compact spaces. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf property, every cover by Gδ sets has a continuum-sized subcollection whose union is Gδ-dense