Dimension theory

Pasynkov, Boris A.; Fedorchuk, V. V.; Филиппов, В. В.

doi:10.1007/bf01091964

Cited by 3 publications

(3 citation statements)

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“…Note now that Hausdorff metric locally coincides with Fedorchuk one [11]. Recall also that the latter metric is inner, thus the estimate L n for Lipschitz constant of c : C(X) → X is globally true for the space of n-nets endowed with Fedorchuk metric.…”

Section: 1mentioning

confidence: 94%

Properties of Two Selections in Metric Spaces of Nonpositive Curvature

Ivanshin

2008

Asian-European J. Math.

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Section: 1mentioning

confidence: 94%

Properties of Two Selections in Metric Spaces of Nonpositive Curvature

Ivanshin

2008

Asian-European J. Math.

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“…At present, there exist various solutions to this problem [4][5][6][7][8][9][10][11]. It is concerned with characterizing dimension functions by small sets of conditions-axioms on given classes of topological spaces.…”

Section: 12mentioning

confidence: 99%

“…Let us remind the reader that S ~ = I ~ for ~ < r (as usual, here I = [0, 1]). At present, there exist various solutions to this problem [4][5][6][7][8][9][10][11]. In [3], Zarelua proved that if X is a completely paracompact space that admits a decomposing mapping into the Hilbert cube I ~, then…”

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confidence: 99%

Dimension axioms and decomposing mappings

Chatyrko¹

1999

J Math Sci

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515.12An axiomatic definition of the covering dimension dim in the c/ass of all (closed) subsets of finitedimensional cubes is given relative to decomposing mappings. An axiomatic definition of the possible transfinite extension of this dimension in the class of all (closed) subsets of the Smirnov compacta is suggested. Bibliography: 15 titles.Below, finite dimension always means finite covering dimension (dim). This paper gives an axiomatic definition of the Lebesgue covering dimension in the class of all closed (arbitrary) subsets of the finite-dimensional cubes I n, n = 1,2,..., relative to decomposing mappings, and an axiomatic definition of the possible transfinite extension zws [1] of this dimension in the class of all closed (arbitrary) subsets of the Smirnov compacta S a, a < wl [2]. Let us remind the reader that S ~ = I ~ for ~ < r (as usual, here I = [0, 1]). A mapping f: X --4 Y is called decomposing [3] if for any point x E X and any of its neighborhoods O, there exists a neighborhood U of fx whose preimage is decomposed into the disjoint union of open subsets V 1 and V 2 such that z E V 1 C O. In [3], Zarelua proved that if X is a completely paracompact space that admits a decomposing mapping into the Hilbert cube I ~, thenwhere zwX = rain{r: there exists a decomposing mapping f: X --+ F}. In particular, inequality (,) holds for separable metric spaces. Let S a, a < wl, be the Smirnov compacta [2]. Then (see [1]) zws X = min{a: there exists a decomposing mapping f: X ~ S'~}. If there is no such a, then we put zws X = co (zwsX is not defined). Let us remind the reader (see [2]) that the class of all (closed) subsets of finite-dimensional cubes I '~, n = 1,2 .... , coincides with the class of all finite-dimensional (compact) second-countable spaces, and the class of all (closed) subsets of the Smirnov compacta S ~ coincides with the class of all (compact) second-countable spaces for which the zws function is defined [1].The axiomatics problem is an old problem of dimension theory. It is concerned with characterizing dimension functions by small sets of conditions-axioms on given classes of topological spaces. At present, there exist various solutions to this problem [4][5][6][7][8][9][10][11]. In particular, Alexandroff [4] suggested a system of axioms defining the Lebesgue covering dimension in the class of all finite-dimensional metrizable compact spaces. In [5], Shchepin gave an axiomatic definition of the covering dimension in the class of all finite-dimensional metrizable spaces by strengthening one of Alexandroff's axioms. Henderson [6] defined a possible transfinite extension of the covering dimension, the D dimension, in the class of metrizable spaces and formulated axioms specifying the D dimension within the class of all possible transfinite extensions of the covering dimension. We suggest axiomatic definitions of the Lebesgue dimension that differ from the Alexandroff and Shchepin axiomatic.s, and the function zws does not coincide with the D dimension even in the class of compact spaces. Al...

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Слабо Бесконечномерные Пространства

Fedorchuk¹

2007

Uspekhi Mat. Nauk

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Обзор посвящен двум сериям новых классов пространств, помежуточных между классом слабо бесконечномерных в смысле П. С. Александрова пространств и классом C-пространств, а именно классам m-C-пространств и w-m-C-пространств, m = 2, 3,. .. , ∞. Классы 2-C и w-2-C совпадают с классом слабо бесконечномерных пространств, а компактные ∞-C-пространства-это в точности C-компакты Хэйвера. На эти классы распространяются основные результаты теории слабо бесконечномерных пространств, включая классификацию посредством трансфинитных лебеговых размерностей и индексов Лузина-Серпинского. Слабые m-Cпространства характеризуются посредством существенных отображений в m-компакты Хендерсона. Доказывается существование наследственно m-сильно бесконечномерных пространств. Библиография: 102 названия.

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Dimension theory

Cited by 3 publications

References 205 publications

Properties of Two Selections in Metric Spaces of Nonpositive Curvature

Properties of Two Selections in Metric Spaces of Nonpositive Curvature

Dimension axioms and decomposing mappings

Слабо Бесконечномерные Пространства

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