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1

Convergence in measure through compactifications

Wajch E.

Czasopismo Techniczne. Nauki Podstawowe

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2016

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Y. 113, iss. 1-NP

137--145

EN

For a metrizable space X, concepts of metric convergence in measure and of functional convergence in measure of sequences of measurable mappings taking their values in X are introduced and applied to a comparison of compactifications of X.

PL

Dla przestrzeni metryzowalnej X, wprowadza się pojęcia metrycznej zbieżności według miary i funkcyjnej zbieżności według miary ciągów odwzorowań mierzalnych, przyjmujących swe wartości w X oraz stosuje się te pojęcia do porównywania uzwarceń przestrzeni X.

2

Countable compact scattered T2 spaces and weak forms of AC

Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2006

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Vol. 54, no 1

75-84

EN

We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T2 topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T2 space is scattered iff it is metrizable. (3) If the real line R can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T2 space is scattered. (4) It is not provable in ZF+¬AC that there exists a countable compact T2 space which is dense-in-itself.

3

Regular covers and metrization

Lin S.

Bulletin of the Polish Academy of Sciences. Mathematics

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2002

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Vol. 50, no 4

427-432

EN

In this paper it is showed that a regular and k-space with a regular k-network is metrizable, which generalized related results of A. Archangielskiĭ, H. W. Martin, M. Sakai, K. Tamano and Y. Yajima.

4

Metrizable weak barrelledness and dimension

Saxon S., Sanchez Ruiz L.

Bulletin of the Polish Academy of Sciences. Mathematics

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2001

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Vol. 49, no 2

97-101

EN

The least infinite-dimensionality for Frechet spaces is c (Mazur), for metrizable barrelled spaces, b (Saxon and Sanchez Ruiz, 1996). For metrizable spaces with the yet weaker inductive property, it is the dimension Aleph[1] of the space chi spanned by any Aleph[1] scalar sequences of the form [(1, x, x^2, x^3, . . .)]. (A locally convex space is inductive if it is the inductive limit of each increasing covering sequence of subspaces). Indeed, chi is a non-barrelled subspace of the Frechet space omega, where the fundamental theorem of algebra at once proves density, dimension and inductivity. Moreover, if each |x| < 1, geometric series then put chi inside the Banach space [l^1], where the identity principle similarly proves the normable case.

5

Uniform properties and hyperspaces of metrizable spaces

Costantini C., Vitolo P.

Journal of Applied Analysis

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1999

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Vol. 5, nr 2

187-196

EN

Two equivalent metrics can be compared, with respect to their uniform properties, in several different ways. We present some of them, and then use one of these conditions to characterize which metrics on a space induce the same lower Hausdorff topology on the hyperspace. Finally, we focus our attention to complete metrics.