Tree duplicates, $G_δ$ -diagonals and Gruenhage spaces

preprint2011arXivOpen access

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.

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Tree duplicates, $G_δ$ -diagonals and Gruenhage spaces

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Tree duplicates, GδG_δGδ​-diagonals and Gruenhage spaces

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Tree duplicates, $G_δ$ -diagonals and Gruenhage spaces