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# On the Stone-Weierstrass Theorem for the Strict and Superstrict Topologies

R. G. Haydon
Proceedings of the American Mathematical Society
Vol. 59, No. 2 (Sep., 1976), pp. 273-278
DOI: 10.2307/2041483
Stable URL: http://www.jstor.org/stable/2041483
Page Count: 6
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## Abstract

Sentilles has introduced topologies $\beta_0, \beta$ and $\beta_1$ on the space $C_b(S)$ of all bounded, continuous, real-valued functions on the completely regular space $S$, which yield as dual spaces the three important spaces of measures, $M_t(S), M_\tau(S)$ and $M_\sigma(S)$, respectively. A number of authors have proved a Stone-Weierstrass theorem for $\beta_0$, the coarsest of the three topologies. In this paper, it is shown that the superstrict topology $\beta_1$ does not obey the Stone-Weierstrass theorem, except perhaps when $\beta_1 = \beta$. Examples are then given to show that the situation for $\beta$ itself is rather complicated.

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