What is Connectivity?

TL;DR

This work develops a unified, point-free taxonomy of connectivity by introducing connectivity lattices and chainmails, bridging point-set and point-free notions. The central tool is the connectivity adjunction, with left and right adjoints encoding joins of mail-connected subsets and the decomposition into connected components, respectively; special cases such as typical, Serra, and kernel connectivities are analyzed. A landmark result is the one-to-one correspondence between absolute connectivity lattices and chainmails, establishing absolute connectivity as a universal, canonical form of connectivity. The framework generalizes classical connectivities (graphs, topological path-connectivity, frames) and clarifies how ambient structure and decomposition interact, with implications for both theory and applications in digital imagery and beyond.

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.

Figures (7)

• Figure 1: Venn diagram visualization of the notion of mail-connectedness in the poset of subsets of a given set. The set of regions marked by the circles is mail-connected, because any two regions can be linked by a sequence of overlapping regions.
• Figure 2: Illustration showing how certain unions of connected regions cannot be cleanly parted without cleanly parting at least one of them
• Figure 3: All connected chainmails having at most 7 elements. Lightly shaded cells represent chainmails that are rooted trees (Example \ref{['exaG']}), except those that are chains. Darkly shaded cells represent those chainmails that are complete lattices.
• Figure 4: Number of mail-connected (single-component) chainmails having $n$ elements (the graph is logarithmic)
• Figure 5: Lattices showing the distinction between the properties \ref{['cond:E1']} and \ref{['cond:E2']}

Theorems & Definitions (80)

• Definition 1
• Theorem 2
• proof
• Example 3
• Theorem 4
• Example 5
• Example 6
• Example 7
• Theorem 8
• Definition 9