The classical Banach-Mazur game is directly related to the Baire property and the property of being a productively Baire space. In this work, we discuss two variations of this classic game that are even more related to these properties.

We characterize the countable dimensionality of metric spaces in terms of a game-theoretic version of a covering property of Haver, and conjecture a game-theoretic equivalent of Haver's covering property for metric spaces.

In[2], the class of compact and extremally disconnected spaces were studied using several investigative tools such as filters, graphs, functions, multifunctions and subsets of the space. These different approaches of investigation produced significant characterizations and properties of this important class of spaces. In our previous paper, "generalized continuous fuctions" we introduced three forms of generalized continuous functions by studying the class of \(u\)-continuous functions of Joseph, Kwack and Nayar [1] using the concepts of an \(\alpha\)-set of Njastad [3]. The generalized continuous forms introduced there are: \(\alpha\)\(u\)-continuous, semi-\(\alpha u\)-continuous and strongly \(u\)-continuous functions. In the present study we investigate the class of compact and extremally disconnected spaces using these generalized continuous functions.

REFERENCES

[1]. J. E. Joseph, M. H. Kwack, On S-closed spaces, Proc. Amer. Math. Soc. 80(1980), no. 2, 341-348.
[2]. B. M. P. Nayar, Compact and extremally disconnected spaces, Fund. Math. Vol 74 (1972), 233-254.
[3]. O. Njastad, On some classes of nearly open sets, International Journal of Mathematics and Mathematical Science, Volume 2004, No 20, April 2004, pp. 1047-1056.

A space \(X\) is called \(Y\)-rigid if any continuous map \(f:X\rightarrow Y\) is constant. For each cardinal \(\kappa\) we construct an infinite \(\kappa\)-bounded (and hence countably compact) regular space \(R_{\kappa}\) such that \(R_{\kappa}\) is \(Y\)-rigid for any \(T_1\) space \(Y\) of pseudo-character \(\leq\kappa\). This result resolves two problems posted by Tzannes in Open Problems from Topology Proceedings and extends results of Ciesielski and Wojciechowski, and Herrlich. Also, we discuss (consistent) examples of regular separable countab\-ly compact \(\mathbb{R}\)-rigid spaces. In particular, we construct a consistent example of regular separable first-countable countably compact \(\mathbb{R}\)-rigid space. Finally, we construct a ZFC example of regular separable sequentially compact space which is not Tychonoff. The latter example resolves a problem of Banakh, Bardyla and Ravsky. The results are contained in joint works with Osipov and Zdomskyy.

Given a completely metrizable space \(X\), let \(\mathfrak{par}(X)\) denote the smallest possible size of a partition of \(X\) into Polish spaces, and \(\mathfrak{cov}(X)\) the smallest possible size of a covering of \(X\) with Polish spaces. Observe that \(\mathfrak{cov}(X) \leq \mathfrak{par}(X)\) for every \(X\), because every partition of \(X\) is also a covering.

I will outline a proof that it is consistent relative to a huge cardinal for the strict inequality \(\mathfrak{cov}(X) < \mathfrak{par}(X)\) to hold for some completely metrizable space \(X\). Using large cardinals is necessary for obtaining this strict inequality, because if \(\mathfrak{cov}(X) < \mathfrak{par}(X)\) for any completely metrizable \(X\), then \(0^\dagger\) exists.

Motivated by results of Juhász and van Mill, the weak tightness \(wt(X)\) of a space \(X\) was introduced by Carlson with the property \(wt(X)\leq t(X)\) where \(t(X)\) is the tightness of \(X\). The cardinal function \(wt(X)\) encodes tightness-like properties of a space that prove sufficient to replace \(t(X)\) with \(wt(X)\) in certain cardinal inequalities. For example, it was shown that if \(X\) is Hausdorff then \(|X|\leq 2^{L(X)wt(X)\psi(X)}\), giving an improvement of a result of Arhangel'skiĭ and Šapirovskiĭ. In 2006 De la Vega proved that \(|X|\leq 2^{t(X)}\) for any homogeneous compactum. In the countably tight case this result was extended in a result of Juhász and van Mill. We introduce the notion of an S-free sequence and show that \(|X|\leq 2^{wt(X)\pi\chi(X)}\) for any homogeneous compactum \(X\). As \(\pi\chi(X)\leq t(X)\) for compacta, this is a general refinement of De la Vega's Theorem. In the case where the cardinal invariants are countable, this represents a variation of the result of Juhász and van Mill. We show further that \(w(X)\leq 2^{wt(X)}\) for a homogeneous compactum \(X\). However, whether \(|X|\leq 2^{wt(X)}\) or \(|X|\leq 2^{\pi\chi(X)}\) for such spaces remain open questions.

We discuss selection principles and games involving hyperspaces and demonstrate that \(k\)-covers of the space of compact subsets with the Vietoris topology can be refined to covers consisting of a certain kind of open set. From this, we derive implications for selection games involving \(k\)-covers. We also apply these techniques to extend results relating \(omega\)-covers of a space to open covers of the finite powers of that space.

We address several questions of Feng, Gruenhage, and Shen which arose from Michael's theory of continuous selections from countable spaces. This theory is concerned with the following general question: given a map from a space \(Y\) to the subsets of another space \(X\), under what conditions is it possible to find a continuous selection?

We construct an example of a space which is \((\omega+1)\)-selective but not \(\mathbb{Q}\)-selective from \(\mathfrak{d}=\omega_1\), and an \((\omega+1)\)-selective space which is not selective for a \(P\)-point ultrafilter from the assumption of \(\mathsf{CH}\). We also produce \(\mathsf{ZFC}\) examples of Fr\'echet spaces where countable subsets are first countable which are not \((\omega+1)\)-selective.

We continue the investigation of the possible dichotomy raised in a question by Istvan Juhasz: Is a compact space first countable if and only if it contains no converging \(\omega_1\)-sequences. An \(\omega_1\)-sequence is converging in a compact space if it contains a unique complete accumulation point. We prove that the question is not decided by the axiom \(\mbox{MA} + \lnot \mbox{CH}\). In particular PFA implies that the dichotomy does hold.

We'll discuss (scattered) compact spaces with a \(P\)-base for some poset \(P\). We show that any compact space with an \(\omega^\omega\)-base is metrizable and any scattered compact space with an \(\omega^\omega\)-base is countable under the assumption \(\omega_1< \mathfrak{b}\). Using forcing, we also prove that in a model of \(\omega_1<\mathfrak{b}\), there is a non-first-countable compact space with a \(P\)-base for some poset \(P\) with calibre~\(\omega_1\).

A non-metrizable manifold is an object satisfying the definition of a manifold, minus paracompactness (so, it is a connected topological space such that each point has an open neighbourhood homeomorphic to \(\mathbb R^n\) for some \(n\in\mathbb N\) — for objects satisfying this definition, paracompactness is equivalent to metrizability). An old theorem of Nyikos, known as the Bagpipe Theorem, characterizes those non-metrizable manifolds that are bagpipes — a connected sum of a compact manifold plus finitely many "long pipes". In joint work with Nicholas Vlamis, we started a study of the space of ends of nonmetrizable manifolds — the space of ends is the remainder of the Freudenthal compactification, and is widely used in recent research in metrizable manifolds — and use this machinery to find a new characterization of bagpipes, as well as a characterization of what we call "generalized bagpipes" — type I manifolds that can be written as the connected sum of a metrizable manifold plus countably many long pipes ("type I" refers to a technical condition only satisfied by nonmetrizable manifolds that are especially well-behaved). In further work along with Vlamis and Mathieu Baillif, we proved that the "type I" requirement cannot be removed from our characterization.

Bonanzinga and Matveev showed that a Mrówka \(\Psi\)-space is strongly star-Menger if and only if the size of the associated almost disjoint family is smaller than the dominating number and, it is strongly star-Hurewicz if and only if the size of the associated almost disjoint family is smaller than the bounding number.

In this talk we introduce definitions that will allow us to reproduce this result in a more general setting. In particular, it includes the selective and absolute versions of these star selection principles and some selective versions of property \((a)\).

It is shown that, assuming the Continuum Hypothesis, every compact Hausdorff space of weight at most \(\mathfrak{c}\) is a remainder in a soft compactification of \(\mathbb{N}\).

We also exhibit an example of a compact space of weight \(\aleph_1\) - hence a remainder in some compactification of \(\mathbb{N}\) - for which it is consistent that is not the remainder in a soft compactification of \(\mathbb{N}\).

In this talk we will generalize work of Kočinac et. al. (2005) and Li (2016) which connects selective separability in the Fell and Vietoris hyperspace topologies to Rothberger-like principles in the corresponding ground space. We make use of and expand on a general game translation tool kit. This unifies and simplifies the methods of Kočinac and Li.

L. B. Treybig showed that if \(X\) and \(Y\) are infinite Hausdorff spaces such that \(X\times Y\) is the continuous image of a compact linearly ordered topological space, then both \(X\) and \(Y\) are metrizable. Later, G. I. Čertanov showed that this theorem holds when we replace a compact linearly ordered space by a countably compact GO space. In this talk, we shall give an outline of a totally different proof of this theorem by using countable elementary submodels.

Given \(n<k<\omega\leq\lambda\), let \(X(n,k,\lambda)\) denote the subspace of the product space \(2^{[\lambda]^n}\) consisting of all \(n\)-uniform hypergraphs on \(\lambda\) that contain no copy of \([k]^n\). We show that \(X(n,k,\lambda)\cong 2^\lambda\) if \(\lambda<\omega_n\) but \(X(n,k,\lambda)\not\cong X(n',k',\lambda)\) if \(\lambda\geq\omega_n\) and \(n<n'<k'<\omega\).

The topological property we use in our proof is the existence of a certain kind of limited-information winning strategy for II in an infinite-length team game with \(n\) players per team.

The most interesting related open problem is, is the space \(X(2,3,\omega_2)\) of triangle-free graphs on \(\omega_2\) homeomorphic to the space \(X(2,4,\omega_2)\) of \(K_4\)-free graphs on \(\omega_2\)?

It is known that the Ellentuck space, which is forcing equivalent to the Boolean algebra \(P(\omega)/\)Fin forces a selective ultrafilter. The Ellentuck space is the prototypical example of a Ramsey space. The connection between Ramsey spaces, ultrafilters, and ideals has been explored in different ways. Ramsey spaces theory has shown to be crucial to investigate Tukey order, Karetov order, and combinatorial properties. This is why we investigate which ideals are related to a Ramsey space in the same sense that the ideal Fin is related to the Ellentuck space. In this talk, we present some results obtained.

The informal term "sprats" is used here for 2-1 closed Hausdorff preimages of \(\omega_1\). These spaces are locally compact, countably compact, first countable, and noncompact.

There is not enough room in a sprat for more than two disjoint copies of \(\omega_1\), and examples are given using \(\clubsuit\) and its weakenings that do not have any copies. But it is also consistent that there must be at least one.

Theorem 1. It is consistent, both with and without CH, that every first countable, countably compact Hausdorff space is either compact or contains a copy of \(\omega_1\).

Balogh showed (1987) the "without" part holds under the PFA, while Eisworth and the speaker showed (2005) that the statement is consistent with CH.

The following results will be the focus of the talk. The first two use only the usual (ZFC) axioms.

Theorem 2. If a sprat does not contain two disjoint copies of \(\omega_1\), then it is normal.

Example 1. There is a non-normal sprat.

The following example uses an axiom that is compatible with a very strong form of MA+not-CH, and shows how much of the power of PFA is used in Theorem 1.

Example 2. If \(\mho_2\), then there is a sprat in which every uncountable subset has the fibers (i.e., point inverses) over a stationary set in its closure.

No sprat with this property can contain a copy of \(\omega_1\). Under the somewhat stronger axiom Club Guessing (which is still compatible with a strong form of MA+not-CH) one can improve "stationary set" to "club subset".

In the 80's, both Abraham-Rudin-Shelah and Todorčević, propose an axiom extending Ramsey Theorem to the uncountable case: the Open Coloring Axiom (OCA). Due to an example given by Sierpiński, it is not possible to have a full generalization of of Ramsey Theorem, so Topology plays an important roll here. In this talk we will investigate what happens to a weakening of these axioms (the Clopen Coloring Axiom) and how it differs from OCA.

Gruenhage and Ma connected Baireness of function spaces with combinatorial properties of objects called moving off families. In this talk these combinatorial properties are are featured, and connected with among other things, a version of Ramsey's theorem.

We present a new set-theoretic axiom we call the Invariant Ideal Axiom and discuss how it can be used to study convergent sequences in sequential and almost sequential groups.

We characterize when the countable power of a Corson compactum has a dense metrizable subspace and construct consistent examples of Corson compacta whose countable power does not have a dense metrizable subspace.

'You can't teach an old dog new tricks,' a popular saying goes. However, can you use old proofs in new contexts? In this talk, I will present results on the existence of long arithmetic progressions in a class of generalised Thue-Morse words. Moreover, I will show how the arguments are inspired by van der Waerden's now classic proof from 1927 of the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers.

The property of being a normal space can be naturally weakened by requiring that pairs of certain types of closed sets can be separated (e.g., Shchepin defined a space to {\em near-normal} if any pair of regular closed sets can be separated by disjoint open sets). We consider a number of different such weakenings in the context \(\Psi\)-spaces over almost disjoint families of subsets of \(\omega\).

During the talk we consider various kinds of Ramsey-type theorems. Bergelson and Hindman investigated finite colorings of the complete graph \([\mathbb{N}]^2\) with vertices in natural numbers, involving an algebraic structure of \(\mathbb{N}\). It follows from their result that for each finite coloring of \([\mathbb{N}]^2\), there are pairwise disjoint sets \(F_1, F_2, \dotsc\) such that each set \(F_n\) contains an arithmetic progression of length \(n\) and all edges between vertices from different sets \(F_n\) have the same color. Colorings of graphs appear also in the context of combinatorial covering properties. Scheepers proved that some of these properties (including Menger and Rothberger) can be characterized using finite colorings of complete graphs whose vertices are open sets of spaces.

The aim of the talk is to present a theorem that captures many results in a similar spirit. To this end, we use topological games and some special ultrafilters in the Stone–Čech compactification of semigroups.

We prove that, for any cofinally Polish space \(X\), every locally finite family of non-empty open subsets of \(X\) is countable. It is also established that Lindelöf domain representable spaces are cofinally Polish and domain representability coincides with subcompactness in the class of \(\sigma\)-compact spaces. It turns out that, for a topological group \(G\) whose space has the Lindelöf \(\Sigma\)-property, the space \(G\) is domain representable if and only if it is Čech-complete. Our results solve several published open questions.

Let \(\mathcal{A}\subseteq\mathbb{R}^\mathbb{R}\) and \(\lambda\) be a cardinal. We say that \(\mathcal{A}\) is \(\lambda\)-lineable if there exists a linear subspace of \(\mathbb{R}^\mathbb{R}\) of dimension \(\lambda\) contained in \(\mathcal{A}\cup\{0\}\). We obtain new lineability results for several families \(\mathcal{A}\subseteq\mathbb{R}^\mathbb{R}\) that are defined using various properties of continuous functions. Such families are usually referred to as families of "Darboux-like functions''. Many of those results are proved in ZFC, but some require additional axioms. For example, we show that assuming the Martin's Axiom, the family of functions that are everywhere surjective, have Strong Cantor Intermediate Value Property, and are not connectivity functions is \(\mathfrak{c}^{+}\)-lineable.

Erdős space \(\mathfrak{E}\) and complete Erdős space \(\mathfrak{E}_c\) have been previously shown to have topological characterizations. In this talk, we provide a topological characterization of the topological space \(\mathbb{Q}\times \mathfrak{E}_c\), where \(\mathbb{Q}\) is the space of rational numbers. As a corollary, we show that the Vietoris hyperspace of finite sets \(\mathcal{F}(\mathfrak{E}_c)\) is homeomorphic to \(\mathbb{Q}\times \mathfrak{E}_c\). We also characterize the factors of \(\mathbb{Q}\times \mathfrak{E}_c\). An interesting open question that is left open is whether \(\sigma \mathfrak{E}_c^\omega\), the \(\sigma\)-product of countably many copies of \(\mathfrak{E}_c\), is homeomorphic to \(\mathbb{Q}\times \mathfrak{E}_c\).

The talk will be devoted to weaker versions of the covering property \(\gamma\) of Gerlits and Nagy, obtained by introducing ideals as parameters. Under CH all of these properties are strictly weaker than being \(\gamma\). As a consequence we get the result stated in the title, thus answering a question of M. Sakai.