Abstract
In Chapter 2 we introduced topological spaces to handle problems of convergence that metric spaces could not. Nevertheless, everyone would rather work with a metric space if they could. The reason is that the metric, a real-valued function, allows us to analyze these spaces using what we know about the real numbers. That is why they are so important in real analysis. We present here some of the more arcane results of the theory. A good source for some of this lesser known material is Kuratowski [149]. Many of these results are the work of Polish mathematicians in the 1920’s and 1930’s. For this reason, a complete separable metric space is called a Polish space.
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A collection F of real-valued functions on a set Ω is called a function space if F under the pointwise operations is a vector lattice. That is, if f, g ∈ F and α ∈ R, then the functions f + g, αf, |f|, f ∨ g, and f ∧ g also belong to F.
In the terminology of Section 7.4, U P (X) is a closed Riesz subspace of С b (Х), and is also an AM-space with unit the constant function one.
This number 6 is known as a Lebesgue number of the cover.
One way of constructing (by induction) such a partition is as follows. Start with N1 = {1,3,5,...} and assume that Nk has been selected so that N\Nk = {n1, n2, nз,...} is countably infinite, where n 1 < n 2 < ... . To complete the inductive argument put Nk+1 = {n1, n3, n5,...}.
We could, if we wished, adopt the conventions that d(x, Ø) = ∞ for every x, so that ρ d (A, Ø) = ∞ for nonempty A and ρ d (Ø, Ø) = 0. This would append the empty set as an isolated point of F. We do not do so however.
The topological lim sup and lim inf of a sequence should be distinguished from the set theoretic lim sup and lim inf of a sequence {E n } defined by
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© 1994 Springer-Verlag Berlin Heidelberg
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Aliprantis, C.D., Border, K.C. (1994). Metrizable spaces. In: Infinite Dimensional Analysis. Studies in Economic Theory, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03004-2_3
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DOI: https://doi.org/10.1007/978-3-662-03004-2_3
Publisher Name: Springer, Berlin, Heidelberg
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