منابع مشابه
Generalizations of Banach-hausdorff Limits
In a recent paper [l],1 W. F. Eberlein introduced the notion of Banach-Hausdorff limits. We employ throughout m to denote the space of all real bounded sequences [2, pp. 11 and 34]. The BanachHausdorff limits are real-valued functionals L(x), defined over m, which are Banach limits [2, p. 34], i.e., which satisfy the four conditions (i) L(ax+by)=aL(x)+bL(y) (a, b real), (ii)L(l) = l, (iii) L(x)...
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Let A ⊆ R be a set definable in an o-minimal expansion S of the real field, A′ ⊆ R be its projection, and assume that the non-empty fibers Aa ⊆ R n are compact for all a ∈ A′ and uniformly bounded. If L is the Hausdorff limit of a sequence of fibers Aai , we give an upper-bound for the Betti numbers bk(L) in terms of definable sets explicitly constructed from a generic fiber Aa. In particular, ...
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We give a geometric proof of the following well-established theorem for o-minimal expansions of the real field: the Hausdorff limits of a compact, definable family of sets are definable. While previous proofs of this fact relied on the model-theoretic compactness theorem, our proof explicitely describes the family of all Hausdorff limits in terms of the original family.
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1950
ISSN: 0002-9939
DOI: 10.2307/2032296