Tree duplicates, $G_\delta$-diagonals and Gruenhage spaces

TL;DR

This paper constructs a specific example of a locally compact, scattered Hausdorff space with a $ ext{G}_ ext{delta}$-diagonal, answering a longstanding question and exploring implications for Banach space smoothness.

Contribution

It provides a ZFC example of a non-Gruenhage space with a $ ext{G}_ ext{delta}$-diagonal and shows that $C_0(D)$ admits a $C^ ext{infty}$-smooth bump function.

Findings

Constructed a non-Gruenhage space with a $ ext{G}_ ext{delta}$-diagonal.

Demonstrated $C_0(D)$ admits a $C^ ext{infty}$-smooth bump function.

Answered a question in the study of convex norms on Banach spaces.

Abstract

We present an example in ZFC of a locally compact, scattered Hausdorff non-Gruenhage space $D$ having a $\G_delta$-diagonal. This answers a question posed by Orihuela, Troyanski and the author in a study of strictly convex norms on Banach spaces. In addition, we show that the Banach space of continuous functions $C_0(D)$ admits a $C^\infty$-smooth bump function.