This is an announcement for the paper "Tree duplicates, $G_\delta$-diagonals and Gruenhage spaces" by Richard J. Smith.

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.

Archive classification: math.FA math.GN

Submitted from: richard.smith@ucd.ie

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1102.0982

or

http://arXiv.org/abs/1102.0982