arXiv subjects
Publications and source records attributed to Steven Clontz.
A standard introductory result is that Hausdorff spaces have the property US, that is, each convergent sequence has a unique limit. This paper explores several existing and new characterizations of separation axioms that are strictly weaker than $T_2$ but strictly stronger than US.
Recent advances in computing have changed not only the nature of mathematical computation, but mathematical proof and inquiry itself. While artificial intelligence and formalized mathematics have been the major topics of this conversation, this paper explores another class of tools for advancing mathematics research: databases of mathematical objects that enable semantic search. In addition to defining and exploring examples of these tools, we illustrate a particular line of research that was inspired and enabled by one such database.
We present a comprehensive report on the relationships between variations of the Menger and Rothberger selection properties with respect to $\omega$-covers and $k$-covers in the most general topological setting and address the finite productivity of some of these properties. We collect various examples that separate certain properties and we carefully identify which separation axioms simplify aspects of these properties. We finish with a consolidated list of open questions focused on topological examples.
Several weakenings of the $T_2$ property for topological spaces, including $k$-Hausdorff, $KC$, weakly Hausdorff, semi-Hausdorff, $RC$, and $US$, have been studied by mathematicians. Here we provide a complete survey of how these properties do or do not relate to one another, including several new results to fill in the gaps in the existing literature, motivated by the use of a community-maintained database of topological spaces and their properties.
Matthew Baker investigated, in previous work, an elegant, infinite-length game that may be used to study subsets of real numbers. We present two accessible examples of how an important technique from set theory, or a different technique from infinite game theory, may be used to answer Baker's question on whether this game provides a precise characterization for countable subsets of real numbers, and we connect this game to the well-studied Banach-Mazur game from topology.
Let $X$ be separable metrizable, and let $f\subseteq X^2$ be a non-trivial relation on $X$. For a given partial order $(P,\leq)$, the Mahavier product $M(X,f,P)\subseteq X^P$ (also known as a generalized inverse limit) collects functions such that $x(p)\in f(x(q))$ for all $p\leq q$. Clontz and Varagona previously showed for well orders $P$ that $M(X,f,P)$ is separable metrizable exactly when $P$ is countable and $f$ satisfies condition $Γ$; we extend this result to hold for all partial orders.
Two selection games from the literature, $G_c(\mathcal O,\mathcal O)$ and $G_1(\mathcal O_{zd},\mathcal O)$, are known to characterize countable dimension among certain spaces. This paper studies their perfect- and limited-information strategies, and investigates issues related to non-equivalent characterizations of zero-dimensionality for spaces that are not both separable and metrizable. To relate results on zero-dimensional and finite-dimensional spaces, a generalization of Telgársky's proof that the point-open and finite-open games are equivalent is demonstrated.
Arhangelskii's properties $α_2$ and $α_4$ defined for convergent sequences may be characterized in terms of Scheeper's selection principles. We generalize these results to hold for more general collections and consider these results in terms of selection games.
An open question of Gruenhage asks if all strategically selectively separable spaces are Markov selectively separable, a game-theoretic statement known to hold for countable spaces. As a corollary of a result by Berner and Juh$\acute{a}$sz, we note that the strong version of this statement, where the second player is restricted to selecting single points in the rather than finite subsets, holds for all $T_3$ spaces without isolated points. Continuing this investigation, we also consider games related to selective sequential separability, and demonstrate results analogous to those for selective separability. In particular, strong selective sequential separability in the presence of the Ramsey property may be reduced to a weaker condition on a countable sequentially dense subset. Additionally, $γ$- and $ω$- covering properties on $X$ are shown to be equivalent to corresponding sequential properties on $C_p(X)$. A strengthening of the Ramsey property is also introduced, which is still equivalent to $α_2$ and $α_4$ in the context of $C_p(X)$.
Often, a given selection game studied in the literature has a known dual game. In dual games, a winning strategy for a player in either game may be used to create a winning strategy for the opponent in the dual. For example, the Rothberger selection game involving open covers is dual to the point-open game. This extends to a general theorem: if $\{\operatorname{ran}{f}:f\in\mathbf C(\mathcal R)\}$ is coinitial in $\mathcal A$ with respect to $\subseteq$, where $\mathbf C(\mathcal R)=\{f\in(\bigcup\mathcal R)^{\mathcal R}:R\in\mathcal R\Rightarrow f(R)\in R\}$ collects the choice functions on the set $\mathcal R$, then $G_1(\mathcal A,\mathcal B)$ and $G_1(\mathcal R,\neg\mathcal B)$ are dual selection games.
We relate the property of discrete selectivity and its corresponding game, both recently introduced by V.V. Tkachuck, to a variety of selection principles and point picking games. In particular we show that player II can win the discrete selection game on \(C_p(X)\) if and only if player II can win a variant of the point open game on \(X\). We also show that the existence of limited information strategies in the discrete selection game on \(C_p(X)\) for either player are equivalent to other well-known topological properties.
Clearly, a generalized inverse limit of metrizable spaces indexed by $\mathbb N$ is metrizable, as it is a subspace of a countable product of metrizable spaces. The authors previously showed that all idempotent, upper semi-continuous, surjective, continuum-valued bonding functions on $[0,1]$ (besides the identity) satisfy a certain Condition $Γ$; it follows that only in trivial cases can a generalized inverse limit of copies of ([0,1]) indexed by an uncountable ordinal be metrizable. The authors show that Condition $Γ$ is in fact guaranteed by much weaker criteria, proving a more general metrizability theorem for certain Mahavier Products.
By adapting techniques of Arhangel'skii, Barman, and Dow, we may equate the existence of perfect-information, Markov, and tactical strategies between two interesting selection games. These results shed some light on Gruenhage's question asking whether all strategically selectively separable spaces are Markov selectively separable.
A relation $f\subseteq X^2$ satisfies condition $Γ$ if there exist distinct $x,y\in X$ with $\langle x,x\rangle ,\langle x,y\rangle ,\langle y,y\rangle \in f$. The authors improve a previous result by characterizing nontrivial idempotent bonding relations on compact Hausdorff spaces as those satisfying condition $Γ$.