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Maps with dimensionally restricted fibers

100%
Valov V.
Colloquium Mathematicum
|
2011
|
tom 123
|
nr 2
239-248
EN
We prove that if f: X → Y is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class S of spaces, then there exists an $F_{σ}$-set A ⊂ X such that A ∈ S and $dim f^{-1}(y)∖A = 0$ for all y ∈ Y. Here, S can be one of the following classes: (i) {M: e-dim M ≤ K} for some CW-complex K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = {M: dim M ≤ n}, then dim f ∆ g ≤ 0 for almost all $g ∈ C(X,𝕀^{n+1})$.
2
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General position properties in fiberwise geometric topology

68%
Banakh T., Valov V.
Instytut Matematyczny Polskiej Akademii Nauk
EN
General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish $LC^{n-1}$-space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding theorem which says that any n-dimensional compact metric space embeds into the (2n+1)-dimensional Euclidean space $ℝ^{2n+1}$. A parametric version of this result was recently proved by B. Pasynkov: any n-dimensional map p: K → M between metrizable compacta with dim M = m embeds into the projection $pr_{M}: M × ℝ^{2n+1+m} → M$ in the sense that there is an embedding $e: K → M × ℝ^{2n+1+m}$ with $pr_{M} ∘ e = p$. This feature of $ℝ^{2n+1+m}$ can be derived from the fact that the space $ℝ^{2n+1+m}$ satisfies the general position property $m - \overline{DD}ⁿ = m - \overline{DD}^{{n,n}}$, which is a particular case of the 3-parameter general position property $m - \overline{DD}^{{n,k}}$ introduced and studied in this paper. We shall give convenient "arithmetic" tools for establishing the $m - \overline{DD}^{{n,k}}$-property and on this base obtain simple proofs of some classical and recent results on (fiber) embeddings. In particular, the Pasynkov theorem mentioned above, as well as the results of P. Bowers and Y. Sternfeld on embedding into a product of dendrites, follow from our general approach. Moreover, the arithmetic of the $m - \overline{DD}^{{n,k}}$-properties established in our paper generalizes some results of W. Mitchell, R. Daverman and D. Halverson. The paper consists of two parts. In the first part we survey the principal results proved in this paper and discuss their applications and interplay with existing results in this area. The second part contains the proofs of the principal results announced in the first part.
3
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Classical-type characterizations of non-metrizable ANE(n)-spaces

64%
Gutev V. G., Valov V.
Fundamenta Mathematicae
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1994
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tom 145
|
nr 3
243-259
EN
The Kuratowski-Dugundji theorem that a metrizable space is an absolute (neighborhood) extensor in dimension n iff it is $LC^{n-1} & C^{n-1}$ (resp., $LC^{n-1}$) is extended to a class of non-metrizable absolute (neighborhood) extensors in dimension n. On this base, several facts concerning metrizable extensors are established for non-metrizable ones.
4
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Spaces with fibered approximation property in dimension n

64%
Banakh T., Valov V.
Open Mathematics
|
2010
|
tom 8
|
nr 3
411-420
EN
A metric space M is said to have the fibered approximation property in dimension n (briefly, M ∈ FAP(n)) if for any ɛ > 0, m ≥ 0 and any map g: $$ \mathbb{I} $$ m × $$ \mathbb{I} $$ n → M there exists a map g′: $$ \mathbb{I} $$ m × $$ \mathbb{I} $$ n → M such that g′ is ɛ-homotopic to g and dim g′ ({z} × $$ \mathbb{I} $$ n) ≤ n for all z ∈ $$ \mathbb{I} $$ m. The class of spaces having the FAP(n)-property is investigated in this paper. The main theorems are applied to obtain generalizations of some results due to Uspenskij [11] and Tuncali-Valov [10].
5
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Open maps having the Bula property

64%
Gutev V., Valov V.
Fundamenta Mathematicae
|
2009
|
tom 205
|
nr 2
91-104
EN
An open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed sets F₀,F₁ ⊂ X with f(F₀) = Y = f(F₁), provided all fibers of f are infinite and C*-embedded in X. Applications are given to the existence of "disjoint" usco multiselections of set-valued l.s.c. mappings defined on paracompact C-spaces, and to special type of factorizations of open continuous maps from metrizable spaces onto paracompact C-spaces. This settles several open questions.
6
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On dimensionally restricted maps

64%
Tuncali H., Valov V.
Fundamenta Mathematicae
|
2002
|
tom 175
|
nr 1
35-52
EN
Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → 𝕀ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,𝕀ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an $F_{σ}$-subset $A_{k}$ of X such that $dim A_{k} ≤ k$ and the restriction $f|(X∖A_{k})$ is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij about extensional dimension is also established.
7
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Skeletally Dugundji spaces

51%
Kucharski A., Plewik S., Valov V.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1949-1959
EN
We introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. Our main result states that the following conditions are equivalent for a given space X: (i) X is skeletally Dugundji; (ii) every compactification of X is co-absolute to a Dugundji space; (iii) every C*-embedding of the absolute p(X) in another space is strongly π-regular; (iv) X has a multiplicative lattice in the sense of Shchepin [Shchepin E.V., Topology of limit spaces with uncountable inverse spectra, Uspekhi Mat. Nauk, 1976, 31(5), 191–226 (in Russian)] consisting of skeletal maps.
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