Introduction To Typed Topological Space

TL;DR

This paper introduces typed topological spaces where open sets are assigned types from a lattice, enabling new definitions of neighborhoods, closure, and connectedness, and incorporates statistical semantics for points.

Contribution

It proposes a novel framework of typed topological spaces with type-based definitions and statistical semantics, expanding traditional topology concepts.

Findings

Typed topological spaces allow flexible neighborhood and closure definitions.

Statistical characteristics can be integrated into topological semantics.

The approach avoids the issue of singleton sets being both closed and open.

Abstract

The concept of typed topological space is introduced, for which open sets in a topology on a finite set will be assigned types (from lattice). The neighborhood system of a point, the closure and the connectedness can be defined according to chain of types, which effectively avoids the situation when most singletons are closed an open. Furthermore, statistics can be used to provide semantics of points with statistical characteristics.