Tree duplicates, $G_{\delta }$-diagonals and Gruenhage 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.

References in corpus (2)

• Strictly convex norms and topology
• Gruenhage compacta and strictly convex dual norms