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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
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2041483
Page Count: 6
You can always find the topics here!Topics: Topology, Topological theorems, Mathematics, Induced substructures, Completely regular spaces, Separable spaces, Mathematical theorems, Analytics, Topological spaces, Topological compactness
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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.
Proceedings of the American Mathematical Society © 1976 American Mathematical Society