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Normal Vietoris implies compactness: a short proof

G. Di Maio, E. Meccariello, Somashekhar Naimpally (2004)

Czechoslovak Mathematical Journal

One of the most celebrated results in the theory of hyperspaces says that if the Vietoris topology on the family of all nonempty closed subsets of a given space is normal, then the space is compact (Ivanova-Keesling-Velichko). The known proofs use cardinality arguments and are long. In this paper we present a short proof using known results concerning Hausdorff uniformities.

Note on countable unions of Corson countably compact spaces

Ondřej F. K. Kalenda (2004)

Commentationes Mathematicae Universitatis Carolinae

We show that a compact space K has a dense set of G δ points if it can be covered by countably many Corson countably compact spaces. If these Corson countably compact spaces may be chosen to be dense in K , then K is even Corson.

On binary coproducts of frames

Xiangdong Chen (1992)

Commentationes Mathematicae Universitatis Carolinae

The structure of binary coproducts in the category of frames is analyzed, and the results are then applied widely in the study of compactness, local compactness (continuous frames), separatedness, pushouts and closed frame homomorphisms.

On certain compact topological spaces.

Manuel Valdivia (1997)

Revista Matemática de la Universidad Complutense de Madrid

A compact topological space K is in the class A if it is homeomorphic to a subspace H of [0,1]I, for some set of indexes I, such that, if L is the subset of H consisting of all {xi : i C I} with xi=0 except for a countable number of i's, then L is dense in H. In this paper we show that the class A of compact spaces is not stable under continuous maps. This solves a problem posed by Deville, Godefroy and Zizler.

On character and chain conditions in images of products

Murray Bell (1998)

Fundamenta Mathematicae

A scadic space is a Hausdorff continuous image of a product of compact scattered spaces. We complete a theorem begun by G. Chertanov that will establish that for each scadic space X, χ(X) = w(X). A ξ-adic space is a Hausdorff continuous image of a product of compact ordinal spaces. We introduce an either-or chain condition called Property R λ ' which we show is satisfied by all ξ-adic spaces. Whereas Property R λ ' is productive, we show that a weaker (but more natural) Property R λ is not productive. Polyadic...

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