A weakly Stegall space that is not a Stegall space

Moors, Warren; Somasundaram, Sivajah

doi:10.1090/S0002-9939-02-06717-5

A weakly Stegall space that is not a Stegall space
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by Warren B. Moors and Sivajah Somasundaram

Proc. Amer. Math. Soc. 131 (2003), 647-654

DOI: https://doi.org/10.1090/S0002-9939-02-06717-5

Published electronically: June 27, 2002

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Abstract:

A topological space $X$ is said to belong to the class of Stegall (weakly Stegall) spaces if for every Baire (complete metric) space $B$ and minimal usco $\varphi :B\rightarrow 2^{X}$, $\varphi$ is single-valued at some point of $B$. In this paper we show that under some additional set-theoretic assumptions that are equiconsistent with the existence of a measurable cardinal there is a Banach space $X$ whose dual, equipped with the weak$^*$ topology, is in the class of weakly Stegall spaces but not in the class of Stegall spaces. This paper also contains an example of a compact space $K$ such that $K$ belongs to the class of weakly Stegall spaces but $(C(K)^*,\mathrm {weak}^*)$ does not.

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