Continuous functions that are locally constant on dense sets. (English) Zbl 0916.46019

Rudin, Mary Ellen; Rudin, Walter

J. Funct. Anal. 133, No. 1, 129-137 (1995).

For a space \(X\), let \(E_0(X)\) be the family of all continuous real-valued functions on \(X\) that are locally constant on a dense (open) subset of \(X\). It was a question of S. Sidney whether \(E_0(X)\) always separates points of \(X\) for a compact Hausdorff space \(X\). The authors present an example of a path-connected compact Hausdorff space \(X\) which answers Sidney’s question in the negative.

Reviewer: Michael G.Tkachenko (México)

MSC:

46E15

Banach spaces of continuous, differentiable or analytic functions

Keywords:

locally constant on a dense (open) subset; separates points; path-connected compact Hausdorff space