What is Connectivity?

TL;DR

This paper develops a comprehensive taxonomy of connectivity concepts across various mathematical structures, unifying ideas from graph theory, topology, and lattice theory.

Contribution

It introduces a unified framework for understanding connectivity in diverse space-like structures inspired by poset embeddings.

Findings

Provides a taxonomy covering all standard notions of connectivity

Unifies concepts from graphs, hypergraphs, topology, and frames

Offers a foundational perspective for future research in connectivity

Abstract

In this paper, we explore a taxonomy of connectivity for space-like structures. It is inspired by isolating posets of connected pieces of a space and examining its embedding in the ambient space. The taxonomy includes in its scope all standard notions of connectivity in point-set and point-free contexts, such as connectivity in graphs and hypergraphs (as well as k-connectivity in graphs), connectivity and path-connectivity in topology, and connectivity of elements in a frame.