The geometric realization of a simplicial Hausdorff space is Hausdorff

TL;DR

This paper proves that the thin geometric realization of a simplicial Hausdorff space retains the Hausdorff property, confirming a longstanding claim by Graeme Segal about simplicial k-spaces.

Contribution

It establishes that the geometric realization of a simplicial Hausdorff space is Hausdorff, resolving a well-known conjecture in topology.

Findings

The geometric realization of a simplicial Hausdorff space is Hausdorff.

Confirms Segal's claim about the realization of simplicial k-spaces.

Provides a rigorous proof of the Hausdorff property preservation.

Abstract

It is shown that the thin geometric realization of a simplicial Hausdorff space is Hausdorff. This proves a famous claim by Graeme Segal that the thin geometric realisation of a simplicial k-space is a k-space.