The geometric realization of a simplicial Hausdorff space is Hausdorff

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.

💡 Research Summary

The paper addresses a long‑standing claim made by Graeme Segal that the thin geometric realization of a simplicial k‑space is itself a k‑space. The authors focus on the Hausdorff case, proving that if each level Xₙ of a simplicial space X· is a Hausdorff space, then its thin geometric realization |X·|ₜ is also Hausdorff. The thin realization is defined as the quotient of the disjoint union ⨆ₙ Xₙ×Δⁿ by the equivalence relation generated by the usual face and degeneracy identifications (x, dᵢt)∼(dᵢx, t) and (x, sᵢt)∼(sᵢx, t). The central difficulty is to show that this quotient does not collapse distinct points in a way that destroys the separation property.

The authors first establish that each equivalence class under this relation is finite and closed in the pre‑quotient space. This follows from the continuity of the face and degeneracy maps and the Hausdorff nature of each Xₙ, which guarantees that images of closed sets remain closed. Consequently, the quotient map is a closed map on each class, a crucial ingredient for preserving Hausdorffness.

Next, they construct separating neighbourhoods for any two distinct points