Using a factorization theorem due to Pasynkov we provide a short proof of the existence and density of parametric Bing and Krasinkiewicz maps. In particular, the following corollary is established: Let $f\colon X\to Y$ be a surjective map between paracompact spaces such that all fibers $f^{-1}(y)$, $y\in Y$, are compact and there exists a map $g\colon X\to\mathbb I^{\aleph_0}$ embedding each $f^{-1}(y)$ into $\mathbb I^{\aleph_0}$. Then for every $n\geq 1$ the space $C^*(X,\mathbb R^n)$ of all bounded continuous functions with the uniform convergence topology contains a dense set of maps $g$ such that any restriction $g|f^{-1}(y)$, $y\in Y$, is a Bing and Krasinkiewicz map.
All spaces in the paper are assumed to be paracompact and all maps continuous. All maps from X to M are denoted by C(X, M). Usually, C(X, M) will carry either the uniform convergence topology or the source limitation topology. When X is compact, these two topologies coincide. Unless stated otherwise, a space (resp., compactum) means a metrizable space (resp., compactum).
In this paper we provide another approach to prove results concerning parametric Bing and Krasinkiewicz maps. The approach is based on Pasynkov’s technique developed in [10] and [11].
Bing maps and Krasinkiewicz maps have been extensively studied recent years (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [13], [14]). Recall that a map f between compact spaces is said to be a Bing map [4] provided all fibers of f are Bing spaces. Here, a compactum is a Bing space if each of its subcontinua is indecomposable. Following Krasinkiewicz [2], we say that a space M is a free space if for any compactum X the function space C(X, M) contains a dense subset consisting of Bing maps. The class of free spaces is quite large, it contains all n-dimensional manifolds (n ≥ 1) [2], the unit interval [4], all locally finite polyhedra [12], all manifolds modeled on the Menger cube M n 2n+1 or the Nöbeling space N n 2n+1 [12], as well as all 1-dimensional locally connected continua [12].
Next theorem follows from the proof of [13,Theorem 1.2] where the special case with X and Y being metrizable was established.
Theorem 1.1. Let M be a free ANR-space and f : X → Y be a perfect map with W (f ) ≤ ℵ 0 , where X and Y are paracompact. Then the maps g ∈ C(X, M) such that all restrictions g|f -1 (y), y ∈ Y , are Bing maps form a dense set B ⊂ C(X, M) with respect to the source limitation topology. Moreover, B is G δ provided Y is first countable.
Here, W (f ) ≤ ℵ 0 (see [10]) means that there exists a map g : X → I ℵ 0 such that f △g : X → Y × I ℵ 0 is an embedding. For example [10,Proposition 9.1], W (f ) ≤ ℵ 0 for any closed map f : X → Y such that X is a metrizable space and every fiber f -1 (y), y ∈ Y , is separable.
Although, the arguments from [13] don’t work when the map f in Theorem 1.1 is not perfect or the space M is not ANR, we have the following result: Theorem 1.2. Let X and Y be paracompact spaces and f : X → Y be a map with compact fibers and W (f ) ≤ ℵ 0 . Then for every compact free space M the space C(X, M) equipped with the uniform convergence topology contains a dense subset of maps g such that all restrictions g|f -1 (y), y ∈ Y , are Bing maps.
The second type of results concern Krasinkiewicz maps. A space M is said to be a Krasinkiewicz space [9] if for any compactum X the function space C(X, M) contains a dense subset of Krasinkiewicz maps. Here, a map g : X → M, where X is compact, is said to be Krasinkiewicz [5] if every continuum in X is either contained in a fiber of g or contains a component of a fiber of g. The class of Krasinkiewicz spaces contains all Euclidean manifolds and manifolds modeled on Menger or Nöbeling spaces, all polyhedra (not necessarily compact), as well as all cones with compact bases (see [3], [5], [6], [8], [9]).
Theorem 1.3. Let f : X → Y be a map with compact fibers and W (f ) ≤ ℵ 0 , where X and Y are paracompact spaces. If M is a compact Krasinkiewicz space, then C(X, M) equipped with the uniform convergence topology contains a dense subset of maps g such that all restrictions g|f -1 (y), y ∈ Y , are Krasinkiewicz maps.
Corollary 1.4. Let f : X → Y be a map with compact fibers such that W (f ) ≤ ℵ 0 , where X and Y are paracompact spaces. Then for every n ≥ 1 the space C * (X, R n ) of all bounded continuous functions with the uniform convergence topology contains a dense set of maps g such that any g|f -1 (y), y ∈ Y , is a Bing and Krasinkiewicz map.
Theorem 1.3 was established in [13,Theorem 1.1] for an arbitrary Krasinkiewicz ANR-space M in the case X, Y are metrizable, f is perfect and C(X, M) is equipped with the source limitation topology. Let us note that the proof of [13, Theorem 1.1] provides the following result: Let f : X → Y be a perfect map between paracompact spaces with W (f ) ≤ ℵ 0 , and let M be a Krasinkiewicz ANR-space. Then the maps g ∈ C(X, M) such that all g|f -1 (y), y ∈ Y , are Krasinkiewicz maps form a dense subset of C(X, M) with respect to the source limitation topology. Moreover, this set is G δ if Y is first countable.
Remark. The requirement in Theorems 1.2 -1.3 f to have compact fibers is necessary because of the definition of Bing and Krasinkiewicz maps. If we define a Bing space to be a space such that any its subcontinuum (a connected compactum) is indecomposable, and a Bing map to be a map whose fibers are Bing spaces, then Theorem 1.2 remains valid for any map f with W (f ) ≤ ℵ 0 . Then same remark is true for Theorem 1.3 if by a Krasinkiewicz map we mean any map g : X → M, where X is not necessary compact, such that every continuum in X is either contained in a fibe
This content is AI-processed based on open access ArXiv data.
