Multiple Cylinder of Relations for Finite Spaces and Nerve Theorem for Strong-Good Cover

TL;DR

This paper introduces a generalized relation cylinder for finite T0-spaces and a strong-good cover concept, leading to an enhanced Nerve Theorem that preserves simple homotopy types.

Contribution

It develops the multiple cylinder of relations for finite spaces and extends the Nerve Theorem using strong-good covers, strengthening classical homotopical results.

Findings

Generalized relation cylinder captures complex homotopical structures.

Strong-good covers ensure intersections are collapsible, not just contractible.

Nerve Theorem is strengthened to preserve simple homotopy types.

Abstract

In this paper, we develop the concept of multiple cylinder of relations which is a generalization of the relation cylinder, extending the multiple non-Hausdorff mapping cylinder to sequences of finite T0-spaces linked by a series of relations. This construction is important in capturing complex homotopical structures across chains of finite spaces and, when the relations are induced by maps, it serves as a third space that collapses to two distinct finite spaces. Additionally, we introduce the concept of a strong-good cover for simplicial complexes and finite spaces, char acterized by collapsible (rather than merely contractible) intersections. This leads to a strengthened version of the Nerve Theorem, which we develop for simplicial complexes as well as for finite spaces with strong-good covers, demonstrating that these complexes and spaces and their associated nerves maintain the same simple homotopy type, thereby refining classical results for finite simplicial complexes and finite topological structures.