New characterisations of compact spaces

Program Structures: Some New Characterisations

Locally Compact, Ω1-compact Spaces

This paper is centered on an extremely general problem: Problem. Is it consistent (perhaps modulo large cardinals) that a locally compact space X must be the union of countably many ω-bounded subspaces if every closed discrete subspace of X is countable [in other words, if X is ω1-compact]? A space is ω-bounded if every countable subset has compact closure. This is a strengthening of countable ...

New Examples of Weakly Compact Approximation in Banach Spaces

The Banach space E has the weakly compact approximation property (W.A.P.) if there is C < ∞ so that the identity map IE can be uniformly approximated on any weakly compact subset D ⊂ E by weakly compact operators V on E satisfying ‖V ‖ ≤ C. We show that the spaces N(`, `) of nuclear operators ` → ` have the W.A.P. for 1 < q ≤ p < ∞, but that the Hardy space H does not have the W.A.P.

Spaces of Compact Operators

In this paper we study the structure of the Banach space K(E, F) of all compact linear operators between two Banach spaces E and F. We study three distinct problems: weak compactness in K(E, F), subspaces isomorphic to l~ and complementation of K(E, F) in L(E, F), the space of bounded linear operators. In § 2 we derive a simple characterization of the weakly compact subsets of K(E, F) using a c...

Fuzzifying Strongly Compact Spaces and Fuzzifying Locally Strongly Compact Spaces

In this paper, some characterizations of fuzzifying strong compactness are given, including characterizations in terms of nets and pre -subbases. Several characterizations of locally strong compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained.