arXiv subjects

Gabriel Fernandes

Publications and source records attributed to Gabriel Fernandes.

12 recordsLinked to original sources

arXiv2026

Cardinality Bounds for Hausdorff SDL Spaces

We establish the cardinal inequality $|X|\leq 2^{t(X)H\psi(X)}$ for every Hausdorff SDL space $X$, where $t(X)$ and $H\psi(X)$ denote the tightness and the Hausdorff pseudocharacter of $X$, respectively. Since both invariants are bounded by $\chi(X)$, this yields $|X|\leq 2^{\chi(X)}$. As a consequence, every first-countable Hausdorff strongly cellular--Lindel\"of space has cardinality at most the continuum. These results answer Questions~2.1 and~2.2 of Bella and Spadaro. An intermediate result is a uniform bounded-decomposition property for SDL spaces; in particular, their strict quasi--Lindel\"of number satisfies $\sqL(X)\leq t(X)$.

math.GN↗

arXiv2026

A Borel Concept Class of VC Dimension One with a Non-PAC Consistent Learner in ZFC

The fundamental theorem of statistical learning states that, under suitable measurability assumptions, finite Vapnik--Chervonenkis (VC) dimension guarantees that every proper consistent learning rule is probably approximately correct (PAC). Blumer, Ehrenfeucht, Haussler, and Warmuth showed, assuming the Continuum Hypothesis, that the "well-behavedness" condition of the concept class cannot be omitted: they constructed a concept class of Borel sets of VC dimension one admitting a consistent learning rule that is not PAC. We show that the Continuum Hypothesis is unnecessary. Working in Zermelo--Fraenkel set theory with the Axiom of Choice (ZFC) alone, we construct a concept class of Borel sets on $[0,1]$ of VC dimension one and a proper consistent learning rule that is not PAC. More precisely, for a suitable Borel probability measure and target concept, the rule has true risk one at every sample size on a set of samples of outer probability one. Consequently, finite VC dimension and Borel measurability of the individual concepts do not suffice to guarantee that every proper consistent learning rule is PAC. The result shows, with no need of extra set-theoretical assumptions, that the additional regularity assumption in the fundamental theorem cannot in general be omitted.

math.LO↗

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↗

arXiv2024

Remarks on Halin's end-degree Conjecture

We prove new instances of Halin's end degree conjecture (HC) in ZFC. In particular, we show that there is a proper class of cardinals kappa for which Halin's conjecture holds, answering two questions posed by Geschke, Kurkofka, Melcher, and Pitz (2023). We also investigate the relationship between HC and the Singular Cardinal Hypothesis, deriving consistency strength from failures of the former. Moreover, we verify that Halin's conjecture fails on finite intervals of successors of singular cardinals in Merimovich's model, yielding a new independence result concerning HC.

math.LO↗

arXiv2024

Ray inflations of $\omega_1$-trees and ends of degree $\aleph_1$

We prove the followings result for ray inflations of sparse graphs on $\omega_1$-trees. First, let $T$ and $S$ be pruned $\omega_1$-trees, let $G_T$ be a sparse $T$-graph, and let $G_S$ be a sparse $S$-graph with uniformly finite adhesion. If $T$ is not special, then no subdivision of $G_S\# \mathbb{N}$ is isomorphic to $G_T\# \mathbb{N}$. Second, if $T$ is almost-Suslin and $S$ is special, then $G_T\# \mathbb{N}$ contains no subgraph isomorphic to $G_S\# \mathbb{N}$, for arbitrary choices of the sparse graphs. Consequently, under $\diamondsuit_{\omega_1}$, this gives a counterexample to Halin's end degree conjecture that is not isomorphic to a subdivision of any ray inflation with uniformly finite adhesion. Under $\diamondsuit_{\omega_1}^{*}$, the underlying tree may in addition be chosen almost-Suslin, and the resulting graph contains no subgraph isomorphic to a ray inflation over a special $\omega_1$-tree.

math.LO↗

arXiv2023

On totally Lindelöf spaces

The results in this paper answer three questions asked by (NOBLE, 2019) and give a partial answer to a question asked by (ALSTER, 1988). We prove that every Alster space is totally Lindelof and this gives a new characterization of regular Alster spaces. We construct a non-regular totally Lindelof space that is not Alster and we prove that there exists a Lindelof P-space that is not Frolik.

math.GN↗

arXiv2023

Reducing the Price of Stable Cable Stayed Bridges with CMA-ES

The design of cable-stayed bridges requires the determination of several design variables' values. Civil engineers usually perform this task by hand as an iteration of steps that stops when the engineer is happy with both the cost and maintaining the structural constraints of the solution. The problem's difficulty arises from the fact that changing a variable may affect other variables, meaning that they are not independent, suggesting that we are facing a deceptive landscape. In this work, we compare two approaches to a baseline solution: a Genetic Algorithm and a CMA-ES algorithm. There are two objectives when designing the bridges: minimizing the cost and maintaining the structural constraints in acceptable values to be considered safe. These are conflicting objectives, meaning that decreasing the cost often results in a bridge that is not structurally safe. The results suggest that CMA-ES is a better option for finding good solutions in the search space, beating the baseline with the same amount of evaluations, while the Genetic Algorithm could not. In concrete, the CMA-ES approach is able to design bridges that are cheaper and structurally safe.

cs.NE↗

arXiv2022

Partitions of topological spaces and a new club-like principle

We give a new proof of the following theorem due to W. Weiss and P. Komjath: if $X$ is a regular topological space, with character $ < \mathfrak{b}$ and $X \rightarrow (top ω+ 1)^{1}_ω$, then, for all $α< ω_1$, $X \rightarrow (top α)^{1}_ω$, fixing a gap in the original one. For that we consider a new decomposition of topological spaces. We also define a new combinatorial principle $\clubsuit_{F}$, and use it to prove that it is consistent with $\neg CH$ that $\mathfrak{b}$ is the optimal bound for the character of $X$. In \cite{WeissKomjath}, this was obtained using $\diamondsuit$.

math.GN↗

arXiv2021

Tall cardinals in extender models

Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $λ$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $κ$ is a tall cardinal if and only if it is either a strong cardinal or a measurable limit of strong cardinals.

math.LO↗

arXiv2021

On Local Club Condensation

We obtain results on the condensation principle called local club condensation. We prove that in extender models an equivalence between the failure of local club condensation and subcompact cardinals holds. This gives a characterization of $\square_κ$ in terms of local club condensation in extender models. Assuming $\gch$, given an interval of ordinals $I$ we verify that iterating the forcing defined by Holy-Welch-Wu, we can preserve $\gch$, cardinals and cofinalities and obtain a model where local club condensation holds for every ordinal in $I$ modulo those ordinals which cardinality is a singular cardinal. We prove that if $κ$ is a regular cardinal in an interval $I$, the above iteration provides enough condensation for the combinatorial principle $\Dl_{S}^{*}(Π^{1}_{2})$, and in particular $\diamondsuit(S)$, to hold for any stationary $S \subseteq κ$.

math.LO↗

arXiv2020

Inclusion modulo nonstationary

A classical theorem of Hechler asserts that the structure $\left(ω^ω,\le^*\right)$ is universal in the sense that for any $σ$-directed poset P with no maximal element, there is a ccc forcing extension in which $\left(ω^ω,\le^*\right)$ contains a cofinal order-isomorphic copy of P. In this paper, we prove a consistency result concerning the universality of the higher analogue $\left(κ^κ,\le^S\right)$: Theorem. Assume GCH. For every regular uncountable cardinal $κ$, there is a cofinality-preserving GCH-preserving forcing extension in which for every analytic quasi-order Q over $κ^κ$ and every stationary subset S of $κ$, there is a Lipschitz map reducing Q to $(κ^κ,\le^S)$.

math.LO↗

arXiv2020

Fake reflection

We introduce a generalization of stationary set reflection which we call "filter reflection", and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We prove the independence of filter reflection from ZFC, and present applications of filter reflection to the study of canonical equivalence relations of the higher Cantor and Baire spaces.

math.LO↗