On Weakly Contractible Non-Contractible Finite Topological Spaces of Ten Points

TL;DR

This paper completes the classification of certain finite topological spaces with ten points, showing their contractibility and explicitly describing their structure and collapsibility.

Contribution

It extends previous work by classifying all homotopically trivial non-contractible finite $T_0$-spaces with ten points, detailing their types and contractibility.

Findings

Six spaces with three middle elements up to homeomorphism.

Four spaces with four middle elements.

All ten spaces have contractible order complexes.

Abstract

Cianci and Ottina proved that a homotopically trivial non-contractible finite $T_0$-space cannot have fewer than nine points and classified all such spaces with exactly nine points. The present paper completes the classification for spaces with exactly ten points. No such space exists when the number of middle elements is one or two; this is established by Euler-characteristic arithmetic, beat-point arguments, and an analysis of forced naked edges. For exactly three middle elements there are precisely six spaces up to homeomorphism, forming three explicit types and their order-duals; for exactly four middle elements there are precisely four such spaces. The ten valid spaces are each shown to have a contractible order complex: seven explicit elementary collapse sequences are given, one for each of Types~I through~VII, and the three remaining spaces, the order-duals of Types~I, II, and~III, inherit contractibility from the identity $\mathcal{K}(X^{\mathrm{op}})=\mathcal{K}(X)$ of simplicial complexes, since chains in $X$ and $X^{\mathrm{op}}$ coincide as sets and any collapse sequence for $\mathcal{K}(X)$ is simultaneously one for $\mathcal{K}(X^{\mathrm{op}})$.