Pathologies in cohomology of non-paracompact Hausdorff spaces

TL;DR

This paper constructs a specific non-paracompact Hausdorff space demonstrating that Cech and sheaf cohomology can differ, revealing limitations in classical cohomological tools for such spaces.

Contribution

It provides a concrete example of a non-paracompact Hausdorff space where Cech and sheaf cohomology diverge, and analyzes properties of the sheaf of continuous functions.

Findings

Cech cohomology does not coincide with sheaf cohomology in the constructed space.

The sheaf of continuous functions is neither soft nor acyclic.

The space admits non-numerable principal bundles.

Abstract

We construct a non-paracompact Hausdorff space for which Cech cohomology does not coincide with sheaf cohomology. Moreover, the sheaf of continuous real-valued functions is neither soft nor acyclic, and our space admits non-numerable principal bundles.