# Multi-topological semantics for intuitionistic modal logic

## TL;DR

This paper introduces multi-topological semantics for intuitionistic modal logic, connecting neighborhood models with topological structures to enhance understanding and reasoning in modal logic.

## Contribution

It develops a novel multi-topological framework for intuitionistic modal logic, linking neighborhood models with topological semantics and proposing transformations between them.

## Key findings

- Neighborhood models can be represented as multi-topological spaces.
- Differences between minimal and maximal neighborhoods are key to semantics.
- Transformations between multi-topological spaces and neighborhood structures are possible.

## Abstract

We present three examples of \textit{multi-topological} semantics for intuitionistic modal logic with one modal operator $\Box$ (which behaves in some sense like necessity). We show that it is possible to treat neighborhood models, introduced earlier, as multi-topological. From the neighborhood point of view, our method is based on differences between properties of minimal and maximal neighborhoods. Also we propose transformation of multi-topological spaces into the neighborhood structures. Although our neighborhoods can be considered as bi-relational models, we believe that reasoning in terms of neighborhoods gives better topological intuitions.