Covering by discrete and closed discrete sets

Say that a cardinal number $\kappa$ is \emph{small} relative to the space $X$ if $\kappa <\Delta(X)$, where $\Delta(X)$ is the least cardinality of a non-empty open set in $X$. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire $\sigma$-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.Comment: 12 pages, to appear on Topology and its Application

Performative perspectives on short story collections

The first part of this paper offers a brief theoretical discussion of the short story collection and raises some concerns about the relevance of its historical roots. In a second part, the concepts of performance and performativity are introduced in order to investigate how these concepts can play a relevant role in the theoretical description of the peculiar functioning of the short story collection as a literary form

A generalized integrability problem for G-Structures

Given an $\widetilde n$-dimensional manifold $\widetilde M$ equipped with a $\widetilde G$-structure $\widetilde\pi:\widetilde P\rightarrow \widetilde M$, there is a naturally induced $G$-structure $\pi: P\rightarrow M$ on any submanifold $M\subset\widetilde M$ that satisfies appropriate regularity conditions. We study generalized integrability problems for a given $G$-structure $\pi: P\rightarrow M$, namely the questions of whether it is locally equivalent to induced $G$-structures on regular submanifolds of homogeneous $\widetilde G$-structures $\widetilde\pi:\widetilde P\to \widetilde{H}/\widetilde{K}$. If $\widetilde\pi:\widetilde P\to \widetilde{H}/\widetilde{K}$ is flat $k$-reductive we introduce a sequence of generalized curvatures taking values in appropriate cohomology groups and prove that the vanishing of these curvatures are necessary and sufficient conditions for the solution of the corresponding generalized integrability problems.Comment: 30 pages, v2: improved presentation and results v3: improved presentation, final version to appear in Ann. Mat. Pura App

Short proof of a theorem of Juhasz

We give a simple proof of the increasing strengthening of Arhangel'skii's Theorem. Our proof naturally leads to a refinement of this result of Juh\'asz.Comment: 5 page

The role of Madīna in the emergence of the Mosque-dār al-Imāra combination. A preliminary note

This contribution aims to complete the debates concerning the Mosque of the Prophet in Madīna with a study focused on the dwelling quarters that populated the surroundings of the blessed building over time (622-711). An in-depth analysis of the sources has allowed us to sketch a reliable plan of the ensemble, finally succeeding in demonstrating that the direct link between the mosque and the caliphal residence - conceivably involving a specific ceremonial purpose - must be post-dated to the Marwānid period

On two topological cardinal invariants of an order-theoretic flavour

Noetherian type and Noetherian $\pi$-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the \emph{cellularity}, that is the supremum of the sizes of pairwise disjoint non-empty open sets in a topological space. Building on that analogy, we study the Noetherian $\pi$-type of $\kappa$-Suslin Lines, and we are able to determine it for every $\kappa$ up to the first singular cardinal. We then prove a consequence of Chang's Conjecture for $\aleph_\omega$ regarding the Noetherian type of countably supported box products which generalizes a result of Lajos Soukup. We finish with a connection between PCF theory and the Noetherian type of certain Pixley-Roy hyperspaces

A note on discrete sets

We give several partial positive answers to a question of Juhasz and Szentmiklossy regarding the minimum number of discrete sets required to cover a compact space. We study the relationship between the size of discrete sets, free sequences and their closures with the cardinality of a Hausdorff space, improving known results in the literature.Comment: 14 pages, to appear on Commentationes Mathematicae Universitatis Carolina