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Paul J. Szeptycki

Publications and source records attributed to Paul J. Szeptycki.

12 recordsLinked to original sources
arXiv2024

$Ψ$-Spaces and Semi-Proximality

We discuss the proximal game and semi-proximality in $Ψ$-spaces of almost disjoint families over an infinite countable set and $Ψ$-spaces of ladder systems on $ω_1$. We show that a semi-proximal almost disjoint families must be nowhere MAD, anti-Luzin and characterize semi-proximality for a class of ${\mathbb R}$-embeddable almost disjoint families. We show that a $Ψ$-spaces defined from a uniformizable ladder system is semi-proximal and a $Ψ$-space defined on a $\clubsuit^*$ sequence is not semi-proximal. Thus the existence of non-semi-proximal $Ψ$-space over a ladder system is independent of ZFC.

math.GN↗
arXiv2020

Weak normality properties in $Ψ$-spaces

Almost disjoint families of true cardinality $\mathfrak{c}$ are used to produce an example of a mildly-normal not partly-normal $Ψ$-space and a quasi-normal not almost-normal $Ψ$-space. This is related with a problem posed by Lufti Kalantan where he asks whether there exists a mad family so that the related Mrówka-Isbell space is partly-normal. In addition, a consistent example of a Luzin mad family such that its associated $Ψ$-space is quasi-normal is provided.

math.GN↗
arXiv2019

The effect of forcing axioms on the tightness of the $G_δ$ modification

We show that $\mathsf{PFA}$ implies that the tightness $t(X_δ)$ of the $G_δ$-modification of a Fréchet $α_1$-space $X$ is at most $ω_1$, while $\Box(κ)$ implies that there is a Fréchet $α_1$-space with $G_δ$-tightness equal to $κ$. We use the example constructed from $\Box(κ)$ to show that a local version of the bound $t(X_δ)\le 2^{t(X)L(X)}$ does not hold. We also construct, assuming $\mathsf{MA}$, an example of a Fréchet space whose $G_δ$-tightness is larger than $ω_1$.

math.GN↗
arXiv2019

Selectivity properties of spaces

This paper addresses several questions of Feng, Gruenhage, and Shen which arose from Michael's theory of continuous selections from countable spaces. We construct an example of a space which is $L$-selective but not $\mathbb{Q}$-selective from $\mathfrak{d}=ω_1$, and an $L$-selective space which is not selective for a $P$-point ultrafilter from the assumption of $\mathsf{CH}$. We also produce $\mathsf{ZFC}$ examples of Fréchet spaces where countable subsets are first countable which are not $L$-selective.

math.GN↗
arXiv2019

A 0-dimensional, Lindelöf space that is not strongly D

A topological space $X$ is strongly $D$ if for any neighbourhood assignment $\{U_x:x\in X\}$, there is a $D\subseteq X$ such that $\{U_x:x\in D\}$ covers $X$ and $D$ is locally finite in the topology generated by $\{U_x:x\in X\}$. We prove that $\diamondsuit$ implies that there is an $HFC_w$ space in $2^{ω_1}$ (hence 0-dimensional, Hausdorff and hereditarily Lindelöf) which is not strongly $D$. We also show that any $HFC$ space $X$ is dually discrete and if additionally, countable sets have Menger closure then $X$ is a $D$-space.

math.GN↗
arXiv2017

Locally finite trees and the topological minor relation

A well-known theorem of Nash-Williams shows that the collection of locally finite trees under the topological minor relation results in a BQO. Set theoretically, two very natural questions arise: (1) What is the number $λ$ of topological types of locally finite trees? (2) What are the possible sizes of an equivalence class of locally finite trees? For (1), clearly, $ω\leq λ\leq \mathfrak{c}$ and Matthiesen refined it to $ω_1 \leq λ\leq \mathfrak{c}$. Thus, this question becomes non-trivial when the Continuum Hypothesis is not assumed. In this paper we address both questions by showing that - entirely within ZFC - for a large collection of locally finite trees that includes those with countably many rays: the answer for (1) is $λ= ω_1$, and that for (2) the size of an equivalence class can only be either $1$ or $\mathfrak{c}$.

math.CO↗
arXiv2016

Uniform Powers of Compacta and the Proximal Game

The countable uniform power (or uniform box product) of a uniform space $X$ is a special topology on ${}^ωX$ that lies between the Tychonoff topology and the box topology. We solve an open problem posed by P. Nyikos showing that if $X$ is a compact proximal space then the countable uniform power of $X$ is also proximal (although it is not compact). By recent results of J. R. Bell and G. Gruenhage this implies that the countable uniform power of a Corson compactum is collectionwise normal, countably paracompact and Fréchet-Urysohn. We also give some results about first countability, realcompactness in countable uniform powers of compact spaces and explore questions by P. Nyikos about semi-proximal spaces.

math.GN↗
arXiv2011

A counterexample in the theory of $D$-spaces

Assuming $\diamondsuit$, we construct a $T_2$ example of a hereditarily Lindelöf space of size $ω_1$ which is not a $D$-space. The example has the property that all finite powers are also Lindelöf.

math.GN↗
arXiv2005

On some classes of Lindelöf Sigma-spaces

We consider special subclasses of the class of Lindelöf Sigma-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space $X$ is in the class $LΣ(\leqκ)$ if it admits a cover by compact subspaces of weight $κ$ and a countable network for the cover. We restrict our attention to $κ\leqω$. In the case $κ=ω$, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the $P$-approximable spaces considered by Tkacenko. The case $κ=1$ corresponds to the spaces of countable network weight, but even the case $κ=2$ gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight $\aleph_1$ is in the class $LΣ(\leq ω)$, answering a question of Tkachuk. As well, we study whether certain compact spaces in $LΣ(\leqω)$ have dense metrizable subspaces, partially answering a question of Tkacenko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.

math.GN↗
arXiv1998

Strongly almost disjoint sets and weakly uniform bases

A combinatorial principle CECA is formulated and its equivalence with GCH+ certain weakenings of Box_lambda for singular lambda is proved. CECA is used to show that certain ``almost point- < tau'' families can be refined to point- < tau families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of ``every first countable T_1-space with a weakly uniform base has a point-countable base.''

math.LO↗
arXiv1995

The combinatorics of open covers (II)

We continue to investigate various diagonalization properties for sequences of open covers of separable metrizable spaces introduced in Part I. These properties generalize classical ones of Rothberger, Menger, Hurewicz, and Gerlits-Nagy. In particular, we show that most of the properties introduced in Part I are indeed distinct. We characterize two of the new properties by showing that they are equivalent to saying all finite powers have one of the classical properties above (Hurewicz property in one case and in the Menger property in other). We consider for each property the smallest cardinality of metric space which fails to have that property. In each case this cardinal turns out to equal another well-known cardinal less than the continuum. We also disprove (in ZFC) a conjecture of Hurewicz which is analogous to the Borel conjecture. Finally, we answer several questions from Part I concerning partition properties of covers.

math.LO↗