On the weak homotopy types of small finite spaces

TL;DR

This paper proves that connected finite topological spaces with up to 12 points have the same weak homotopy type as a wedge of spheres, revealing their simplified topological structure.

Contribution

It establishes that small connected finite spaces have a homotopy type equivalent to a wedge of spheres, extending understanding of their topological classification.

Findings

Finite connected spaces with ≤12 points are weakly homotopy equivalent to wedges of spheres.

Order complexes of small finite posets share the same homotopy type as wedges of spheres.

Provides a classification result for the topology of small finite spaces.

Abstract

We show that a connected finite topological space with $12$ or less points has a weak homotopy type of a wedge of spheres. In other words, we show that the order complex of a connected finite poset with $12$ or less points has a homotopy type of a wedge of spheres.