Rogério Augusto dos Santos Fajardo

arXiv2026

Translations between interior preorder structures and coherent neighbourhood systems for intuitionist modal logic

On this article we provide a relation between two inherently different semantic structures for intuitionistic modal logic. We start by recalling the Heyting Algebras, then defining the language and the axioms for the iS4h intuitionistic calculus. We then proceed to analyse two structures discussed on [4] and their relation. Finally, we discuss about a structure defined on [3, Chapter 6], establishing properties in order for it to be sound with iS4h, and finally conclude that it is semantically equivalent to the other two structures defined by [4].

math.LO↗

arXiv2016

Suprema of continuous functions on connected spaces

Let $K$ be a compact Hausdorff space and let $(f_n)_{n\in \N}$ be a pairwise disjoint sequence of continuous functions from $K$ into $[0,1]$. We say that a compact space $L$ \emph{adds supremum} of $(f_n)_{n\in \N}$ in $K$ if there exists a continuous surjection $π:L\longrightarrow K$ such that there exists $sup\{f_n\circπ:n\in \N\}$ in $C(L)$. Moreover, we expect that $L$ preserves suprema of disjoint continuous functions which already existed in $C(K)$. Namely, if $sup\{g_n:n\in \N\}$ exists in $C(K)$, we must have $sup\{g_n\circπ:n\in \N\}$ in $C(L)$. This paper studies the preservation of connectedness in extensions by continuous functions -- a technique developed by Piotr Koszmider to add suprema of continuous functions on Hausdorff connected compact spaces -- proving the following results: (1) If $K$ is a metrizable and locally connected compactum, then any extension of $K$ by continuous functions is connected (but it may be not locally connected). (2) There exists a disconnected extension of a metrizable connected compactum $K$. (3) For any metrizable compactum $K$ there exists a disconnected $L$ which is obtained from $K$ by finitely many extensions by continuous functions.

math.GN↗