arXiv · 2610.25156
This note provides a negative answer to the well-known Reich's problem in fixed point theory.
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Luong Van Nguyen. 2026-10-18. A counterexample to Reich's fixed point problem. https://arxiv.org/abs/2610.25156
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We study topological vector group topologies on the real line generated by Hamel coefficient functions and circle-valued characters. We determine their continuous characters, completions, and isomorphisms, and construct a sequence separating modular convergence from convergence of real-valued observations. We also classify precompact topologies and prove that no greatest nondiscrete topology exists. For the separate problem of a greatest compatible locally quasi-convex topology, we establish transfer and dilation criteria and rule out a family of dyadic counterexample constructions. The existence of a greatest compatible topology remains unresolved.
In this paper, we study Levine equivalence in aura topological spaces. We show that the aura topology is determined by the reachability preorder induced by the scope function and is therefore an Alexandrov topology. We prove that the aura-Levine hull has the explicit representation $K_{\mathfrak a}(A)=\uparrow_{\mathfrak a}A=S_{\mathfrak a}^{\infty}(A)$, and hence two subsets are aura-Levine equivalent if and only if they have the same eventual forward spread. We also describe the quotient structure of the equivalence classes, characterize separation properties through the reachability preorder, classify scope functions that induce the same Levine equivalence, and examine preservation under aura-continuous mappings. For finite aura spaces, we provide an explicit procedure for deciding aura-Levine equivalence using the reflexive-transitive closure of the scope relation.
Let $X$ be a topological space. Galvin's theorem on the Banach-Mazur game states that if $X$ has a $π$-base $\mathcal{B}$ satisfying the following condition: \textit{For any $U \in \mathcal{B}$, there exists a disjoint family $\mathcal{A}_U$ of nonempty open subsets of $U$ such that} \[ |\mathcal{A}_U| \geq |\{ B \in \mathcal{B} : U \subseteq B \}|, \] then, in the Banach-Mazur game on $X$, the second player, denoted NONEMPTY, has a 2-tactic if NONEMPTY has a winning strategy in this game. In this paper, we extend the disjointness condition of this family by replacing it with the condition of being open finite family. With this extension of the family $\mathcal{A}_U$, we showed that, the theorem is also true.