arXiv subjects

Matheus Duzi

Publications and source records attributed to Matheus Duzi.

6 recordsLinked to original sources

arXiv2026

Ray and end spaces: characterizations and classification up to homeomorphism

We provide a combinatorial characterization for pairs of order-theoretic trees with homeomorphic ray spaces, answering an open problem proposed by Kurkofka ad Pitz. This solution is inspired by the introduction of a transfinite topological game, which allows us to characterize not only ray spaces through the existence of winning strategies for one of the players, but also their homeomorphic classes. As applications of these results, we obtain a new topological characterization for graph-theoretic end spaces (thus obtaining yet another solution to a recently solved problem of Diestel), as well as for edge-end spaces and completely ultrametrizable spaces. We also introduce a generalization of the class of ray spaces (which is strict, as witnessed by the Sorgenfrey line). Furthermore, we establish that, for subspaces with cardinality less than continuum of end spaces, the scattered property is equivalent to the property of being, itself, an end space. At last, we determine that ray spaces in a couple of classes fail to have their product with any non-discrete space as a ray space.

math.GN↗

arXiv2025

Categories of Games and their Fra\"iss\'e Theory

Relying on recent generalizations of the Fra\"iss\'e theory to a broader category-theoretic context, we study the class of abstract finite games played between two players and show the existence of an infinitetly countable game which is ultrahomogeneous and universal with respect to said class. Certain peculiarities of our game categories which clash with the usual framework found in the literature then lead us to formulate weaker category-theoretic properties which still yield a universal and ultrahomogeneous Fra\"iss\'e limit, thus further generalizing the categorical framework for a Fra\"iss\'e theory.

math.GM↗

arXiv2025

On covering properties of end and ray spaces

We provide new results on combinatorial characterizations of covering properties in end spaces and ray spaces. In particular, we characterize the Lindel\"of degree, the extent, the Rothberger property, $\sigma$-compactness and the Menger property for ray, end and edge-end spaces. We show that $\sigma$-compactness and the Menger property are equivalent for these spaces, and that they are all $D$-spaces. As an application of some of these characterizations, we are able to provide combinatorial characterizations of graphs with countably many ends and edge-ends.

math.CO↗

arXiv2025

On edge-direction and compact edge-end spaces

Directions of graphs were originally introduced in the study of a cops-and-robbers kind of game, while the study of end spaces has been used to generalize classical graph-theoretical results to infinite graphs, such as Halin's generalization of Menger's theorem. An edge-analogue of end spaces, where finite sets of edges are used instead of vertices as separator agents to form the so-called edge-end space, has been recently used to obtain an edge-analogue of this later result. Inspired by Diestel's correspondence between directions and ends of a graph, we tackle in this paper an edge-analogue of directions, its relation with line graphs, and an edge-analogue of Diestel's correspondence. The results of this study had some implications over edge-end space compactness, which then became a target of inquiry: we thus show an edge-analogue of Diestel's combinatorial characterization for compact end spaces. Non-edge-dominating vertices play an important role in our characterization, which motivated the study of ends and directions using now finite sets of these vertices as separator agents, as done previously for edges, giving rise to other topological spaces associated with graphs. These new direction and end spaces once again motivate an analogue of Diestel's correspondence result, and further generalizations are obtained. All of these constructions define topological space-classes associated with graphs such as edge-end spaces and edge-direction spaces of graphs. The paper organizes these topological space-classes appearing throughout the text with representation results, as it was done by Pitz and Kurkofka, as well as Aurichi, Real and Magalh\~aes J\'unior. Most notably, we show that every compact edge-end space can be represented as the edge-direction space of a connected graph.

math.CO↗

arXiv2024

Infinitely ludic categories

Pursuing a new approach to the study of infinite games in combinatorics, we introduce the categories $\mathbf{Game}_{A}$ and $\mathbf{Game}_{B}$ and improve some classical results concerning topological games related to the duality between covering properties of $X$ and convergence properties of $\mathrm{C}_{\mathrm {p}}(X)$ by establishing the existence and key role of certain natural transformations. We then describe these ludic categories in various equivalent forms, viewing their objects as certain structured trees, presheaves, or metric spaces, and we thereby obtain their arboreal, functorial and metrical appearances. We use their metrical disguise to demonstrate a universality property of the Banach-Mazur game. The various equivalent descriptions come with underlying functors to more familiar categories which help establishing some important properties of the game categories: they are complete, cocomplete, extensive, cartesian closed, and coregular, but neither regular nor locally cartesian closed. We prove that their classes of strong epimorphisms, of regular epimorphisms, and of descent morphisms, are all distinct, and we show that these categories have weak classifiers for strong partial maps. Some of the categorical constructions have interesting game-theoretic interpretations.

math.GN↗

arXiv2019

Topological games of bounded selections

We present a new variation of the classical selection principles $\mathsf{S}_\mathrm{k}(\mathcal A, \mathcal B)$ ($k\in\mathbb N$) and $\mathsf{S}_\mathrm{fin}(\mathcal A, \mathcal B)$ that formally lies between these two properties. As in the case of the classical selection principles, we also obtain a new variation of topological games and discuss how new topological properties may emerge in the specific cases of covering and tightness.

math.GN↗