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𝒫 -approximable compact spaces

Mihail G. Tkachenko (1991)

Commentationes Mathematicae Universitatis Carolinae

For every topological property 𝒫 , we define the class of 𝒫 -approximable spaces which consists of spaces X having a countable closed cover γ such that the “section” X ( x , γ ) = { F γ : x F } has the property 𝒫 for each x X . It is shown that every 𝒫 -approximable compact space has 𝒫 , if 𝒫 is one of the following properties: countable tightness, 0 -scatteredness with respect to character, C -closedness, sequentiality (the last holds under MA or 2 0 < 2 1 ). Metrizable-approximable spaces are studied: every compact space in this class has...

A Cantor set in the plane that is not σ-monotone

Aleš Nekvinda, Ondřej Zindulka (2011)

Fundamenta Mathematicae

A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y) ≤ cd(x,z) for all x < y < z in X, and σ-monotone if it is a countable union of monotone subspaces. A planar set homeomorphic to the Cantor set that is not σ-monotone is constructed and investigated. It follows that there is a metric on a Cantor set that is not σ-monotone. This answers a question raised by the second author.

A continuous operator extending ultrametrics

I. Stasyuk, Edward D. Tymchatyn (2009)

Commentationes Mathematicae Universitatis Carolinae

The problem of continuous simultaneous extension of all continuous partial ultrametrics defined on closed subsets of a compact zero-dimensional metric space was recently solved by E.D. Tymchatyn and M. Zarichnyi and improvements to their result were made by I. Stasyuk. In the current paper we extend these results to complete, bounded, zero-dimensional metric spaces and to both continuous and uniformly continuous partial ultrametrics.

A discrete version of the Brunn-Minkowski inequality and its stability

Michel Bonnefont (2009)

Annales mathématiques Blaise Pascal

In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for metric measure spaces. Our new definition, based only on properties of the distance, allows also us to deal with discrete metric measure spaces. Then we show the stability of our new inequality under convergence of metric measure spaces. This result gives as corollary the stability of the classical Brunn-Minkowski inequality for geodesic spaces. The proof of this stability...

A factorization theorem for the transfinite kernel dimension of metrizable spaces

M. Charalambous (1998)

Fundamenta Mathematicae

We prove a factorization theorem for transfinite kernel dimension in the class of metrizable spaces. Our result in conjunction with Pasynkov's technique implies the existence of a universal element in the class of metrizable spaces of given weight and transfinite kernel dimension, a result known from the work of Luxemburg and Olszewski.

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