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Displaying 401 – 420 of 1977

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Compactness and countable compactness in weak topologies

W. Kirk (1995)

Studia Mathematica

A bounded closed convex set K in a Banach space X is said to have quasi-normal structure if each bounded closed convex subset H of K for which diam(H) > 0 contains a point u for which ∥u-x∥ < diam(H) for each x ∈ H. It is shown that if the convex sets on the unit sphere in X satisfy this condition (which is much weaker than the assumption that convex sets on the unit sphere are separable), then relative to various weak topologies, the unit ball in X is compact whenever it is countably compact....

Compactness in Metric Spaces

Kazuhisa Nakasho, Keiko Narita, Yasunari Shidama (2016)

Formalized Mathematics

In this article, we mainly formalize in Mizar [2] the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness,...

Compactness of Powers of ω

Paolo Lipparini (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary extensions. We also have results involving products of possibly uncountable regular cardinals.

Complete 0 -bounded groups need not be -factorizable

Mihail G. Tkachenko (2001)

Commentationes Mathematicae Universitatis Carolinae

We present an example of a complete 0 -bounded topological group H which is not -factorizable. In addition, every G δ -set in the group H is open, but H is not Lindelöf.

Currently displaying 401 – 420 of 1977