We consider the class of proximal and semi-proximal spaces defined by Jocelyn Bell and introduce a strengthening of this class by examining the proximal game defined on totally bounded uniformities. We also discuss recent results about proximal and semi-proximal spaces. Joint work with Paul Szeptycki.

We describe how to obtain a maximal quotient flow of a flow of a discrete group on an extremally disconnected space when we equip the group with a non-discrete topology. This generalized such description previously done for special types of flows, namely the greatest ambit and the Samuel compactification.

In 1976, Gary Gruenhage introduced what he described as “a simple two-person infinite game,” now sometimes called the neighborhood-point game. Among other things, he used this game to prove results about the preservation of topological properties in products. This infinite game was the inspiration for another game, called the proximal infinite game, which, not coincidentally, has also been used to prove results on products. We will discuss these games, certain modifications, and some results obtained by playing these games. Along the way we highlight parallels with Gruenhage's work.

Barber and Erde asked the following question: if $B$B generates $\mathbb{Z}^n$Zn as an additive group, then must the extremal sets for the isoperimetric inequality on the Cayley graph $(\mathbb{Z}^n,B)$(Zn,B) form a nested family? We answer this question negatively for both the vertex- and edge-isoperimetric inequalities, already when $n=1$n=1. The key is to show that the structure of the cylinder $\mathbb{Z}\times(\mathbb{Z}/k\mathbb{Z})$Z×(Z/kZ) can be mimicked in certain Cayley graphs on $\Z$Z, leading to a phase transition. Based on joint work with Chris Wells.

We show, in a certain specific sense, that both the density and the cardinality of a Hausdorff space are related to the "degree" to which the space is nonregular. It was shown by Sapirovskii that $d(X)\leq\pi\chi(X)^{c(X)}$d(X)≤πχ(X)c(X) for a regular space $X$X and the speaker observed this holds if the space is only quasiregular. We generalize this result to the class of all Hausdorff spaces by introducing the nonquasiregularity degree $nq(X)$nq(X), which is countable when $X$X is quasiregular, and showing $d(X)\leq\pi\chi(X)^{c(X)nq(X)}$d(X)≤πχ(X)c(X)nq(X) for any Hausdorff space $X$X. This demonstrates that the degree to which a space is nonquasiregular has a fundamental and direct connection to its density and, ultimately, its cardinality. Importantly, if $X$X is Hausdorff then $nq(X)$nq(X) is "small" in the sense that $nq(X)\leq\min\{\psi_c(X),L(X),pct(X)\}$nq(X)≤min{ψc​(X),L(X),pct(X)}. This results in a unified proof of both Sapirovskii's density bound for regular spaces and Sun’s bound $\pi\chi(X)^{c(X)\psi_c(X)}$πχ(X)c(X)ψc​(X) for the cardinality of a Hausdorff space $X$X. A consequence is an improved bound for the cardinality of a Hausdorff space. We give an example of a compact, Hausdorff space for which this new bound is a strict improvement over Sun's bound.

We review various applications of strategic translations in topological selection games and also discuss some particular cases where direct applications fail.

We will introduce high dimensional versions of sequential compactness for every ordinal $\alpha<\omega_1$α<ω1​. This will generalize a previous notion introduced by W. Kubis and P. Szeptycki for $\alpha\in\omega$α∈ω. We then extend some known results in the finite case to the infinite case, exhibit some conditions that imply sequential compactness for higher dimensions and analyze the impact of some cardinal invariants in these classes of spaces. We will close with some remarks and applications.

We investigate the situation regarding autohomeomorphisms of $\mathbb{N}^*$N∗, primarily in the Mathias model.

Joint work with V. Valov

We consider uniformly continuous surjections between $C_p(X)$Cp​(X) and $C_p(Y)$Cp​(Y) (resp, $C_p^*(X)$Cp∗​(X) and $C_p^*(Y$Cp∗​(Y)) and show that if $X$X has some dimensional-like properties, then so does $Y$Y. In particular, we prove that if $T:C_p^*(X)\to C_p^*(Y)$T:Cp∗​(X)→Cp∗​(Y) is a continuous linear surjection, then $\dim Y=0$dimY=0 provided $\dim X=0$dimX=0. This provides a partial answer to a question raised by Kawamura-Leiderman.

Directed sets are common in topology and in a variety of contexts. We show that every directed set can be represented, up to Tukey equivalence, by such a topological directed set. in the opposite direction we show that any totally bounded uniformity is Tukey equivalent to $[\kappa]^{<\omega}$[κ]<ω , the collection of all finite subsets of $\kappa$κ, the cofinality of the uniformity - all other Tukey types are ‘rejected’. Some applications of these results will be discussed in the talk. This is a joint work with Paul Gartside.

We will discuss some recent results, including ZFC inequalities, concerning the higher Baire spaces analogues of some of the classical combinatorial cardinal characteristics of the continuum. Of special interest for the talk will be the generalized bounding, splitting, refining and dominating numbers.

In 1969, Arhangel'skiĭ proved that if $X$X is a Hausdorff space, then $|X|\le 2^{\chi(X)L(X)}$∣X∣≤2χ(X)L(X), where $\chi(X)$χ(X) is the character and $L(X)$L(X) is the Lindelöf degree of $X$X. Since then it has been an open question if his inequality is true for every $T_1$T1​-space $X$X. In 2013, we proved that if $X$X is a $T_1$T1​-space, then $|X|\le nh(X)^{\chi(X)L(X)}$∣X∣≤nh(X)χ(X)L(X), where $nh(X)$nh(X) is the non-Hausdorff number of $X$X. In that way we were able to positively answer this question for every $T_1$T1​-space for which $nh(X)\le 2^{\chi(X)L(X)}$nh(X)≤2χ(X)L(X), and, in particular, when $nh(X)$nh(X) is not grater than the cardinality of the continuum. A simple example shows that our inequality is not always true for $T_0$T0​-spaces.

Arhangel'skiĭ and Šapirovskiĭ strengthened Arhangel'skiĭ's inequality in 1974 by showing that if $X$X is a Hausdorff space, then $|X|\le 2^{t(X)\psi(X)L(X)}$∣X∣≤2t(X)ψ(X)L(X), where $t(X)$t(X) is the tightness and $\psi(X)$ψ(X) is the pseudocharacter of $X$X.

In this talk we will show how Arhangel'skiĭ--Šapirovskiĭ's inequality, and therefore, Arhangel'skiĭ's inequality, could be extended to be valid for all topological spaces.

Joint work with Miguel A. Sánchez-Granero and Cristina Martín-Aguado.

In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with respect to some measure defined on the Borel $\sigma$σ-algebra of the space. We give the notion of Darboux sums and lower and upper Riemann integrals of a bounded function when given a measure and a fractal structure. Furthermore, we give the notion of a Riemann-integrable function in this context and prove that each $\mu$μ-measurable function is Riemann-integrable with respect to $\mu$μ. Moreover, if $\mu$μ is the Lebesgue measure, then the Lebesgue integral on a bounded set of $\mathbb{R}^n$Rn meets the Riemann integral with respect to the Lebesgue measure in the context of measures and fractal structures. Finally, we give some examples showing that we can calculate improper integrals and integrals on fractal sets.

We establish relationships between various topological selection games involving the space of minimal cusco maps into the real line and the underlying domain of those maps. These connections occur across different topologies, including the topology of pointwise convergence and the topology of uniform convergence on compacta. Full and limited-information strategies are investigated. The primary games we consider are Rothberger-like games, generalized point-open games, strong fan-tightness games, Tkachuk's closed discrete selection game, and Gruenhage's (W)-games.

We discuss in what circumstances forcing adds new continuous maps. We prove that if $X$X is scattered compact Hausdorff and $Y$Y is discrete, then forcing does not add any continuous maps from $X$X to $Y$Y. On the other hand, if $X$X is not a zero-dimensional scattered pseudocompact space and $Y$Y has more than one point, then ccc forcing adds a continuous map from $X$X to $Y$Y.

Hart and Kunen and, independently, Ríos-Herrejón defined and studied the class $C({\omega}_1)$C(ω1​) of topological spaces $X$X having the property that for every neighborhood assignment $\{U(y) : y \in Y\}${U(y):y∈Y} with $Y \in [X]^{\omega_1}$Y∈[X]ω1​ there is $Z \in [Y]^{\omega_1}$Z∈[Y]ω1​ such that $Z \subset \bigcap \{U(z) : z \in Z\}.$Z⊂⋂{U(z):z∈Z}. It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. We present several independence results concerning the relationships of these two and several other natural classes that are sandwiched between them.

In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in $C({\omega}_1)$C(ω1​) has countable net weight, answering a question that was raised by Hart and Kunen.

These results are joint with L. Soukup and Z. Szentmiklossy.

The class of $L\Sigma(\leq\omega)$LΣ(≤ω)-spaces was introduced in 2006 by Kubiś, Okunev and Szeptycki as a natural refinement of the classical and important notion of Lindelof $\Sigma$Σ-spaces. Compact $L\Sigma(\leq\omega)$LΣ(≤ω)-spaces were considered earlier, under different names, in the works of Tkachuk and Tkachenko in relation to metrizably fibered compacta. In this talk we will present counterexamples to several open questions about compact $L\Sigma(\leq\omega)$LΣ(≤ω)-spaces that are scattered in the literature. Among other things, we refute a conjecture of Kubiś, Okunev and Szeptycki by constructing a separable Rosenthal compactum which is not an $L\Sigma(\leq\omega)$LΣ(≤ω)-space. We also give insight to the structure of first-countable $(K)L\Sigma(\leq\omega)$(K)LΣ(≤ω)-compacta. The talk is based on a joint work with Antonio Aviles.

Joint work with W. D. Burgess

In joint work with Barr and Kennison it was shown that commutative semiprime rings have a $DL$DL-closure and a $2$2-$3$3 closure and that the ring of continuously differentiable real-valued functons is not closed in either sense. Our work is devoted to trying to describe the two closures of this ring. The methods are analytic often using basic ideas from calculus. A useful example sent by Alan Dow is presented.

We will present several new characterizations of the fact that a given compact space $K$K is Corson compact. Some of them will be in terms of embeddings of $K$K in function spaces, another ones in terms of dense subspaces of $C_p(K)$Cp​(K) and even one characterization in terms of embedding $K$K in a topological group.

We say that a subset of the plane is a $\textit{square-catcher}$square-catcher if it contains at least one vertex of each square of the plane. In this talk, motivated by the square peg problem —does every Jordan curve contains the four vertices of a euclidean square?—, we will present some properties of square-catchers. In particular, we will look at minimal elements, with respect to the subset relation, of the family of square-catchers.

Joint work with J. Zapletal

A dynamical ideal consists of a group acting on a set, along with an ideal that is invariant under the group action, and we can use dynamical ideals to obtain models of choiceless set theory. We focus on dynamical ideals where the underlying set is taken to be a topological space and the acting group is the group of homeomorphisms and look at how dynamical properties of the space correspond to fragments of AC in the associated model of set theory, along with particular examples.