arXiv subjects

Rodrigo R. Dias

Publications and source records attributed to Rodrigo R. Dias.

5 recordsLinked to original sources

arXiv2017

On spaces with $σ$-closed-discrete dense sets

The main purpose of this paper is to study \emph{$e$-separable spaces}, originally introduced by Kurepa as $K_0'$ spaces; we call a space $X$ $e$-separable iff $X$ has a dense set which is the union of countably many closed discrete sets. We primarily focus on the behaviour of $e$-separable spaces under products and the cardinal invariants that are naturally related to $e$-separable spaces. Our main results show that the statement "there is a product of at most $\mathfrak c$ many $e$-separable spaces that fails to be $e$-separable'" is equiconsistent with the existence of a weakly compact cardinal.

math.GN↗

arXiv2016

Selective game versions of countable tightness with bounded finite selections

For a topological space $X$ and a point $x \in X$, consider the following game -- related to the property of $X$ being countably tight at $x$. In each inning $n\inω$, the first player chooses a set $A_n$ that clusters at $x$, and then the second player picks a point $a_n\in A_n$; the second player is the winner if and only if $x\in\overline{\{a_n:n\inω\}}$. In this work, we study variations of this game in which the second player is allowed to choose finitely many points per inning rather than one, but in which the number of points they are allowed to choose in each inning has been fixed in advance. Surprisingly, if the number of points allowed per inning is the same throughout the play, then all of the games obtained in this fashion are distinct. We also show that a new game is obtained if the number of points the second player is allowed to pick increases at each inning.

math.GN↗

arXiv2014

Indestructibility of compact spaces

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to $ω_1$-sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba's "$\aleph_1$-Borel Conjecture" is equiconsistent with the existence of an inaccessible cardinal.

math.GN↗

arXiv2014

Selective Games on Binary Relations

We present a unified approach, based on dominating families in binary relations, for the study of topological properties defined in terms of selection principles and the games associated to them.

math.GN↗

arXiv2013

Topological games and Alster spaces

In this paper we study connections between topological games such as Rothberger, Menger and compact-open, and relate these games to properties involving covers by G_δ subsets. The results include: (1) If Two has a winning strategy in the Menger game on a regular space X, then X is an Alster space. (2) If Two has a winning strategy in the Rothberger game on a topological space X, then the G_δ-topology on X is Lindelof. (3) The Menger game and the compact-open game are (consistently) not dual.

math.GN↗