Research related to general topology and its interplay with axiomatic set theory. Organizers: Jocelyn Bell, Christopher Caruvana, Steven Clontz.
A topic of continued interest in set-theoretic topology is the question of which topological properties are preserved under which forcings. In recent work, the speaker and Holshouser have shown that strongly proper forcings preserve a wide variety of covering properties (including Lindelöf), generalizing work of Dow, Iwasa, and Kada. In this talk, we will give an overview of yet further work done by the speaker on this topic. Namely, we discuss how to create forcing axioms for proper posets that preserve a given Lindelöf space. This uses Neeman's two-type side conditions machinery in combination with the earlier work of Gilton and Holshouser.
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Suppose that a topological space $X$ has no uncountable discrete subspace. We discuss if $X$ can obtain an uncountable discrete subspace in forcing extensions. We prove that for any monotonically normal space $X$ which has no uncountable discrete subspace, $X$ can obtain an uncountable discrete subspace in some forcing extension if and only if $X$ is not separable.
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We present internal characterizations for an inverse system of compact Hausdorff spaces that show when its limit will be strongly infinite-dimensional, weakly infinite-dimensional, or have its dimension $n\in\mathbb{N}_{\geq0}$. Our main tool involves lifting the notion of an essential family into a parallel concept for inverse systems. In our presentation we plan to review the definitions of essential family, strong and weak infinite-dimensionality, finite dimensionality, and inverse systems. After doing that, we will state our main results but will not go into any proofs. The published paper with all details appears in *Rad Hazu. Matematičke Znanosti*, v. 29=564 (2025): 299-318.
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We investigate the existence of closed copies of the discrete space $\mathbb{N}$ of natural numbers in powers of the real line, in particular its $\omega_1$-power, that are not $C^\star$-embedded, or that are $C^\star$-embedded but not $C$-embedded. In the case of non-$C^\star$-embedding we find a whole family of new examples, based on Aronszajn trees and lines, and a combinatorial translation of the existence of such copies. In the case of $C^\star$- but not $C$-embedding we complement an earlier consistency result but showing in consistent with any desired cardinal arithmetic that $\mathbb{R}^{\omega_1}$ contains a closed copy of $\mathbb{N}$ that is $C^\star$- but not $C$-embedded.
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A subset $D$ is a discretely dense subset of a space $X$ if every point of $X$ is in the closure of a discrete subset of $D$. The cardinal invariant, $Dd(X)$, was introduced by Juhasz and is the minimum cardinality of a discretely dense subset of $X$. We are reporting on some recent work with Juhasz and van Mill on results that improve upon the, seemingly only, obvious inequalities $d(X)\leq Dd(X)\leq |X|$. We also consider, $Fd(X)$, the free sequence density number.
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In this talk some recent results about cardinal inequalities for topological spaces will be presented and some open questions will be discussed.
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A space $X$ has the disjoint (discrete) shrinking property if for any family $\{U_n: n\in\omega\}$ of non-empty open subsets of $X$ there exists a disjoint (discrete) family $\{V_n: n\in\omega\}$ of non-empty open sets such that $V_n \subset U_n$ for every $n\in\omega$. We present a topological equivalent of the disjoint shrinking property in general spaces and apply it to characterize the disjoint shrinking property in topological groups and locally convex spaces.
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In this talk, we present the concepts of expansive homeomorphisms in the context of quasi-uniform spaces. We continue with our analysis on expansive homeomorphisms by extending the results from quasi-metric spaces to quasi-uniform spaces. It turns out that an expansive homeomorphism on a quasi-uniform space is also an expansive homeomorphism on its induced quasi-uniformity but the converse does not hold in general. We show that if a homeomorphism on a quasi-uniform space is expansive then the quasi-uniform space is always a Kolmogorov space. Moreover, we generalize the concept of expansive measures in the sense of Morales and Sirvent to quasi-uniform spaces point of views.
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We introduce new selection principles involving networks, namely, M-nw-selective, R-nw-selective, and H-nw-selective. These spaces represent a strengthening of both M-separability, R-separability, and H-separability, as well as the Menger, Rothberger, and Hurewicz properties. We also define and investigate two new games: the R-nw-selective game and the M-nw-selective game, which arise naturally from their corresponding selection principles. We give consistent results, and we define trivial R-, H-, and M-nw-selective spaces the ones with countable netweight and cardinality and weight strictly less than $\text{cov}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$, respectively. We establish that spaces with cardinalities greater than $\text{cov}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$ fail to possess the R-, H-, and M-nw-selective properties, respectively. Non-trivial examples, therefore, should eventually have weight greater than or equal to these small cardinals. Using forcing methods, we construct consistent countable non-trivial examples of R-nw-selective and H-nw-selective spaces. Finally, we study relations between nw-selective properties and a strong version of the HFD property.
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A $\Delta$-space is a Tychonoff space with the property that every partition of the space (into arbitrary sets) has a point finite open expansion. M. Reed defined a set of reals with this property to be a $\Delta$-set and was motivated by the characterization of a $\Delta$-set as those sets of reals $X$ for which the Moore plane over $X$ is countably paracompact. Recently, Leiderman and Kąkol characterized $\Delta$-spaces as those $X$ for which the locally convex space $C_p(X)$ is distinguished. I will survey some recent results concerning $\Delta$-spaces and mention a number of open problems.
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