Ordinal products of topological spaces

Chatyrko, Vitalij A.

doi:10.4064/fm-144-2-95-117

Cited by 5 publications

(10 citation statements)

References 9 publications

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“…A transfinite generalization of the natural sum in which the supremum is taken at limit stages has been considered in Chatyrko [7] with applications to topological dimension theory. The particular case of the sum of a sequence of length ω has been used in Wang [25] and Väänänen and Wang [24] with applications to infinitary logics.…”

Section: B) D)]mentioning

confidence: 99%

An Infinite Natural Product

Lipparini

2018

Annales Mathematicae Silesianae

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Abstract. We study a countably infinite iteration of the natural product between ordinals. We present an "effective" way to compute this countable natural product; in the non trivial cases the result depends only on the natural sum of the degrees of the factors, where the degree of a nonzero ordinal is the largest exponent in its Cantor normal form representation. Thus we are able to lift former results about infinitary sums to infinitary products. Finally, we provide an order-theoretical characterization of the infinite natural product; this characterization merges in a nontrivial way a theorem by Carruth describing the natural product of two ordinals and a known description of the ordinal product of a possibly infinite sequence of ordinals.

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Section: B) D)]mentioning

confidence: 99%

An Infinite Natural Product

Lipparini

2018

Annales Mathematicae Silesianae

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“…If we consider the set A in Definition 1.2.2 to be a point, then we get the definition of the transfinite dimension id (see 50,98), and if both sets A and B are considered to be points (compacta), then idp(c Id) (see [50]). …”

Section: Classification Of Infinite-dimensional Spacesmentioning

confidence: 99%

“…In this section, we consider a variant of the definition of the product of an infinite number of topological spaces, namely, the ordinal products [50], for which, as distinct from the Tichonoff product, there exist, in a compact case, nontrivial solutions of a natural extension to an infinite number of factors of p. …”

Section: Ordinal Produetlonmentioning

confidence: 99%

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The dimension problem for the product of infinite-dimensional spaces

Chatyrko¹

1996

J Math Sci

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“…Recall that DS ~ = idS ~ ---a [6,13] for a < wl and idX ) indX [13]. Recall that DS ~ = idS ~ ---a [6,13] for a < wl and idX ) indX [13].…”

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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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Ordinal products of topological spaces

Cited by 5 publications

References 9 publications

An Infinite Natural Product

An Infinite Natural Product

The dimension problem for the product of infinite-dimensional spaces

Dimension axioms and decomposing mappings

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