B.A. Pasynkov’s research while affiliated with Moscow State Forest University and other places

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Publications (34)


Coincidence points in the cases of metric spaces and metric maps
  • Article

December 2015

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39 Reads

Topology and its Applications

Thi Hong Van Nguyen

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B.A. Pasynkov

In the first half of the paper, we are concerned with the problems of existence (and searching) of coincidence points and the common preimage of a closed subset (in particular, a common root) in the case of a finite system of mappings of one metric space to another one. The second half of the paper is devoted to fiberwise variants of Arutyunov's theorem on coincidence points. Obtaining the main results of the paper is based on the use of the class of almost exactly (α,β)-search functionals that is wider than Fomenko's class of (α,β)-search functionals.



Metric and metrizable mappings

July 2013

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49 Reads

Topology and its Applications

A simplified variant of the definition of the completeness for metric mappings (that is closed to the standard definition of the completeness for metric spaces) is obtained. This result is used to construct the completions of metric mappings by a method close to the standard completion method for metric spaces. Relations between completions, fibrewise completions and fibrewise complete extensions of metric mappings are clarified. It is shown that for closed metric mappings all of these extensions coincide (since the completeness of these mappings coincides with their fibrewise completeness). The Lavrentieff theorem (about GδGδ-extensions of homeomorphisms between subsets of metric spaces) is extended to metric mappings. It is proved that a uniformly continuous map-morphism of metric mappings may be extended to a uniformly continuous map-morphism of their completions. The end of the paper contains generalizations of the Nagata–Smirnov and Bing metrization theorems for mappings.


What is a non-metrizable analog of metrizable compacta? (Part I)

April 2012

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10 Reads

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6 Citations

Topology and its Applications

Two classes of compacta were introduced: the class of metrcompacta and more wide class of weak metrcompacta. Both classes are countably productive. The class of weak metrcompacta is a strict subclass of uniform Eberlein compacta. For any cardinal number τ, there exists a rather simple metrcompactum that is a topologically universal element in the class of all weak metrcompacta (and metrcompacta) of weight τ. Every weak metrcompactum (in particular, every metrcompactum) has a 0-dimensional map onto a metrizable compactum and so the dimensions dim, ind, Ind and Δ coincide for all weak metrcompacta (and metrcompacta). Every metrizable space X has a compactification cX that is a metrcompactum with dimX⩽dimcX.


On dimensional properties of topological products

April 2012

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9 Reads

Topology and its Applications

For any n=2,3,…n=2,3,…, there exist a metrizable compactum ΦnΦn and a compactum YnYn such that dimΦn(=indΦn)=dimYn=indYn=n and dim(Φn×Yn)=n+1<2n=ind(Φn×Yn)dim(Φn×Yn)=n+1<2n=ind(Φn×Yn). For any Tychonoff space X and any metrizable space Y, we have:(1) indX×Y⩽indX+dimY and(2) indX×Y⩽indX+indY if, additionally, Y is strongly metrizable (in particular, strongly paracompact, or separable, or compact).


Multistage n-dimensional universal spaces and extensions

March 2010

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6 Reads

Topology and its Applications

Let Iτ be the Tychonoff cube of weight τ⩾ω with a fixed point, στ and Στ be the correspondent σ- and Σ-products in Iτ and στ⊂(Σστ=(στ)ω)⊂Στ. Then for any n∈{0,1,2,…}, there exists a compactum Unτ⊂Iτ of dimension n such that for any Z⊂Iτ of dimension⩽n, there exists a topological embedding of Z in Unτ that maps the intersections of Z with στ, Σστ and Στ to the intersections Uσnτ, UΣσnτ and UΣnτ of Unτ with στ, Σστ and Στ, respectively; Uσnτ, UΣσnτ and UΣnτ are n-dimensional and Uσnτ is σ-compact, UΣσnτ is a Lindelöf Σ-space and UΣnτ is a sequentially compact normal Fréchet–Urysohn space. This theorem (on multistage universal spaces of given dimension and weight) implies multistage extension theorems (in particular, theorems on Corson and Eberlein compactifications) for Tychonoff spaces.




Some Problems in the Dimension Theory of Compacta

December 2007

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8 Reads

This chapter discusses some problems in the dimension theory of compacta. All topological spaces considered in this chapter are assumed to be Tychonoffand called simply spaces; maps mean continuous maps of topological spaces. Almost all problems posed in the chapter concern compact spaces. The chapter defines the dimension Δ of a paracompact space X as: ΔX ≤n if there exists a strongly zero-dimensional paracompact space X0 and a surjective closed map f: X0 →X such that |f-1x| ≤n +1 for any x ∈X. It is known that the three basic dimension functions dim, ind, and Ind coincide for compact metrizable spaces, that is, dim X = ind X = Ind X for any compact metrizable space X. In 1936, Alexandroffasked whether they coincide for arbitrary compact spaces. In 1941, he proved that dim X ≤ ind X for any compact space X. Also Ind X ≤ Ind X for any normal space X and Ind X ≤ ΔX for any paracompact space X. The chapter discusses about on the coincidence of dim, ind, Ind, and Δ for compact spaces. It explains noncoincidence of dim and ind for compact spaces as well as noncoincidence of ind and Ind for compact spaces. A detailed discussion on dimensional properties of topological products is also presented.


Covering dimension of topological products

July 2007

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45 Reads

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10 Citations

Journal of Mathematical Sciences

The first part of the paper is concerned with conditions under which the inequality dim X × Y ≤ dim X + dim Y and similar inequalities for infinite topological products hold. The second part contains examples of spaces such that the sums of their dimensions are smaller than the dimensions of their products.


Citations (12)


... For Lokucievskiȋ's example see also R. Engelking [9, Examples 2.2.14 and 3.1.31]. For more references see[9], V. A. Chatyrko, K. L. Kozlov, B. A. Pasynkov[6],[7], and V. V. Fedorchuk[10]. ...

Reference:

Components and inductive dimensions of compact spaces
On an approach to constructing compacta with different dimensions dim and ind
  • Citing Article
  • January 2000

Topology Proceedings

... In [2], both Theorems 1.1 and 1.3 are generalized to arbitrary compactifications of two Tychonoff spaces. This paper is devoted to an extension of Theorems 1.1 and 1.3 and their generalizations (given in [2]) to the class of WZ-mappings (in particular, closed mappings) from a locally compact Tychonoff space or a k-absolute space to a compact Hausdorff space. ...

On the homeomorphism of spaces and Magill-type theorems
  • Citing Article
  • Full-text available
  • January 2001

... Let's note that in 1974 H. P. Rosenthal [28] gave an internal characterization of Eberlein compacta (see Theorem 2.1 below). In [10,13] we characterized internally the subspaces of Eberlein compacta, as well as the subspaces of Corson compacta, of uniform Eberlein compacta, of n-uniform Eberlein compacta (in the sense of B. A. Pasynkov [25]) and of strong Eberlein compacta. ...

What is a non-metrizable analog of metrizable compacta? (Part I)
  • Citing Article
  • April 2012

Topology and its Applications

... Since e − dim T ≤ e − dim(X × I n ) ≤ L, we can use the factorization theorem for extension dimension (see [24,Theorem 2], [21]) to construct the inverse system S T = {Z α , P m (ξ β α ) : α, β ∈ A} in such a way that e − dim T α ≤ L, α ∈ A. The equalities Z α = P m (f α ) −1 (Y α ) and T α = p −1 α (Z α ) may not be true for some α, but we always have the inclusions ...

A theorem on ?-maps for maps
  • Citing Article
  • October 1984

Russian Mathematical Surveys