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Spaces with fibered approximation property in dimension n

100%

Banakh T. , Valov V.

tom 8

nr 3

411-420

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].

On dimensionally restricted maps

100%

Tuncali H. , Valov V.

tom 175

nr 1

35-52

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.

Classical-type characterizations of non-metrizable ANE(n)-spaces

100%

Gutev V. G. , Valov V.

tom 145

nr 3

243-259

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.