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Normal spaces and the Lusin-Menchoff property

Pavel Pyrih (1997)

Mathematica Bohemica

We study the relation between the Lusin-Menchoff property and the F σ -“semiseparation” property of a fine topology in normal spaces. Three examples of normal topological spaces having the F σ -“semiseparation” property without the Lusin-Menchoff property are given. A positive result is obtained in the countable compact space.

Normal subspaces in products of two ordinals

Nobuyuki Kemoto, Tsugunori Nogura, Kerry Smith, Yukinobu Yajima (1996)

Fundamenta Mathematicae

Let λ be an ordinal number. It is shown that normality, collectionwise normality and shrinking are equivalent for all subspaces of ( λ + 1 ) 2 .

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.

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