Related papers: On Finite-Dimensional Maps II

We prove that if $f\colon X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class of space $\mathrm S$, then there exists an $F_\sigma$-set $A\subset X$ such that $A\in\mathrm S$ and…

Let f:X -> Y be an onto map between compact spaces such that all point-inverses of f are zero-dimensional. Let A be the set of all functions u:X -> I=[0,1] such that $u[f^\leftarrow(y)]$ is zero-dimensional for all y in Y. Do almost all…

Let $f\colon X\to Y$ be a perfect surjective map of metrizable spaces. It is shown that if $Y$ is a $C$-space (resp., $\dim Y\leq n$ and $\dim f\leq m$), then the function space $C(X,\uin^{\infty})$ (resp., $C(X,\uin^{2n+1+m})$) equipped…

Let $M$ be a complete metric $ANR$-space such that for any metric compactum $K$ the function space $C(K,M)$ contains a dense set of Bing (resp., Krasinkiewicz) maps. It is shown that $M$ has the following property: If $f\colon X\to Y$ is a…

We prove that any continuous mapping $f:E\to Y$ on a completely metrizable subspace $E$ of a perfect paracompact space $X$ can be extended to a Lebesgue class one mapping $g:X\to Y$ (i.e. for every open set $V$ in $Y$ the preimage…

If $g$ is a map from a space $X$ into $\mathbb R^m$ and $z\not\in g(X)$, let $P_{2,1,m}(g,z)$ be the set of all lines $\Pi^1\subset\mathbb R^m$ containing $z$ such that $|g^{-1}(\Pi^1)|\geq 2$. We prove that for any $n$-dimensional metric…

Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that $\dim Y\leq m$. It is shown that, for every positive integer $p\geq 1$ there exists a dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$…

Let X and Y be smooth varieties of dimensions n-1 and n over an arbitrary algebraically closed field, f:X-> Y a finite map that is birational onto its image. Suppose that f is curvilinear; that is, at every point of X, the Jacobian has rank…

Let $f\colon X\to Y$ be a perfect map between finite-dimensional metrizable spaces and $p\geq 1$. It is shown that the space $C^*(X,\R^p)$ of all bounded maps from $X$ into $\R^p$ with the source limitation topology contains a dense…

Every open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed subsets of X so that their image by f is Y provided all fibers of f are infinite and C*-embedded in X. Applications are demonstrated for the…

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…

We say that a metrizable space $M$ is a Krasinkiewicz space if any map from a metrizable compactum $X$ into $M$ can be approximated by Krasinkiewicz maps (a map $g\colon X\to M$ is Krasinkiewicz provided every continuum in $X$ is either…

If $g$ is a map from a space $X$ into $\mathbb R^m$ and $q$ is an integer, let $B_{q,d,m}(g)$ be the set of all lines $\Pi^d\subset\mathbb R^m$ such that $|g^{-1}(\Pi^d)|\geq q$. Let also $\mathcal H(q,d,m,k)$ denote the maps $g\colon…

Hurewicz' characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the…

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps $f:X\to Y$ of the class $\alpha$ for wide classes of topological spaces. In particular, we prove that for a topological space $X$…

We prove that if $X$ is a paracompact connected space and $Z=\prod_{s\in S}Z_s$ is a product of a family of equiconnected metrizable spaces endowed with the box topology, then for every Baire-one map $g:X\to Z$ there exists a separately…

The main result is the following. Let $f \colon X \rightarrow Y$ be a continuous mapping of a completely Baire space $X$ onto a hereditary weakly Preiss-Simon regular space $Y$ such that the image of every open subset of $X$ is a resolvable…

We prove that, if the closed unit ball of a normed space $X$ has sufficiently many extreme points, then every mapping $\Phi$ from $X$ into itself with the following property is affine: For any pair of points in $X$, there exists a (not…

This paper concerns various models of ``at-most-$n$-valued maps''. That is, multivalued maps $f:X\multimap Y$ for which $f(x)$ has cardinality at most $n$ for each $x$. We consider 4 classes of such maps which have appeared in the…

We prove that if $X$ is a paracompact space, $Y$ is a metric space and $f:X\to Y$ is a functionally fragmented map, then (i) $f$ is $\sigma$-discrete and functionally $F_\sigma$-measurable; (ii) $f$ is a Baire-one function, if $Y$ is weak…