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Proceedings of the American Mathematical Society

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The minimal support for a continuous functional on a function space

Author: Kazuhiko Morishita
Journal: Proc. Amer. Math. Soc. 114 (1992), 585-587
MSC: Primary 54C05; Secondary 54C30, 54C35
MathSciNet review: 1070528
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Abstract: Let $ {C_p}(X)$ be the function space with the pointwise convergent topology over a Tychonoff space $ X$ and $ \xi $ a continuous real-valued function on $ {C_p}(X)$. A closed subset $ S$ of $ X$ is called a support for $ \xi $ if $ \xi (f) = \xi (g)$ holds for any pair $ (f,g)$ of elements of $ {C_p}(X)$ such that $ {f_{\vert S}} = {g_{\vert S}}$. It is proven that the minimal support for any real-valued continuous function on the space $ {C_p}(X)$ exists.

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  • [A] A. V. Arhangel'skii, Function spaces with the pointwise topology. I, General Topology, Function Spaces and Dimension, Moscow Univ., 1985, pp. 3-66. (Russian)
  • [E] Ryszard Engelking, General topology, PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. MR 0500780

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Keywords: Function space, functional, pointwise convergent topology, support
Article copyright: © Copyright 1992 American Mathematical Society