General Topology

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New submissions for Tue, 14 May 24

[1]  arXiv:2405.07705 [pdf, ps, other]

Title: Set Convergences via bornology

Authors: Yogesh Agarwal, Varun Jindal

Subjects: General Topology (math.GN); Functional Analysis (math.FA)

This paper examines the equivalence between various set convergences, as studied in [7, 13, 22], induced by an arbitrary bornology $\mathcal{S}$ on a metric space $(X,d)$. Specifically, it focuses on the upper parts of the following set convergences: convergence deduced through uniform convergence of distance functionals on $\mathcal{S}$ ($\tau_{\mathcal{S},d}$-convergence); convergence with respect to gap functionals determined by $\mathcal{S}$ ($G_{\mathcal{S},d}$-convergence); and bornological convergence ($\mathcal{S}$-convergence). In particular, we give necessary and sufficient conditions on the structure of the bornology $\mathcal{S}$ for the coincidence of $\tau_{\mathcal{S},d}^+$-convergence with $\mathsf{G}_{\mathcal{S},d}^+$-convergence, as well as $\tau_{\mathcal{S},d}^+$-convergence with $\mathcal{S}^+$-convergence. A characterization for the equivalence of $\tau_{\mathcal{S},d}^+$-convergence and $\mathcal{S}^+$-convergence, in terms of certain convergence of nets, has also been given earlier by Beer, Naimpally, and Rodriguez-Lopez in [13]. To facilitate our study, we first devise new characterizations for $\tau_{\mathcal{S},d}^+$-convergence and $\mathcal{S}^+$-convergence, which we call their miss-type characterizations.

Cross-lists for Tue, 14 May 24

[2]  arXiv:2405.07112 (cross-list from math.LO) [pdf, other]

Title: Definable compactness in o-minimal structures

Authors: Pablo andújar Guerrero

Comments: This is a round-up of the topological content of arXiv:2111.03802

Subjects: Logic (math.LO); General Topology (math.GN)

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various definitions of definable compactness in the literature, including those in terms of definable curves, definable types and definable downward directed families of closed sets.

[3]  arXiv:2405.07114 (cross-list from math.LO) [pdf, ps, other]

Title: Definable separability and second-countability in o-minimal structures

Authors: Pablo Andújar Guerrero

Subjects: Logic (math.LO); General Topology (math.GN)

We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. We do so by introducing first-order characterizations -- definable separability and definable second-countability -- which make sense in a wider model-theoretic context. We prove that within o-minimality these notions have the desired properties, including their equivalence among definable metric spaces, and conjecture a definable version of Urysohn's Metrization Theorem.

Replacements for Tue, 14 May 24

[4]  arXiv:2207.14738 (replaced) [pdf, ps, other]

Title: Relatively Anosov representations via flows II: examples

Authors: Feng Zhu, Andrew Zimmer

Comments: 52 pages. Final version to appear in the Journal of the London Mathematical Society. arXiv admin note: text overlap with arXiv:2207.14737

Subjects: General Topology (math.GN); Differential Geometry (math.DG); Dynamical Systems (math.DS)

[5]  arXiv:2103.10727 (replaced) [pdf, ps, other]

Title: Fixed points of asymptotically nonexpansive mappings with center 0 and applications

Authors: Abdelkader Dehici, Sami Atailia, Najeh Redjel

Subjects: Functional Analysis (math.FA); General Topology (math.GN)

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