The following theorem is proved. Let f: X → Y be a finite-to-one map such that the restriction $f|f^{-1}\left(S\right)$ is an inductively perfect map for every countable compact set S ⊂ Y. Then Y is a countable union of closed subsets $Y_i$ such that every restriction $f|f^{-1}\left(Y_i\right)$ is an inductively perfect map.

It is shown that for every at most k-to-one closed continuous map f from a non-empty n-dimensional metric space X, there exists a closed continuous map g from a zero-dimensional metric space onto X such that the composition f∘g is an at most (n+k)-to-one map. This implies that f is a composition of n+k-1 simple ( = at most two-to-one) closed continuous maps. Stronger conclusions are obtained for maps from Anderson-Choquet spaces and ones that satisfy W. Hurewicz's condition (α). The main tool is...

A space Y is called a free space if for each compactum X the set of maps with hereditarily indecomposable fibers is a dense $G_{\delta }$-subset of C(X,Y), the space of all continuous functions of X to Y. Levin proved that the interval I and the real line ℝ are free. Krasinkiewicz independently proved that each n-dimensional manifold M (n ≥ 1) is free and the product of any space with a free space is free. He also raised a number of questions about the extent of the class of free spaces. In this paper we will...

Let X and Y be two Polish spaces. Functions f,g: X → Y are called equivalent if there exists a bijection φ from X onto itself such that g∘φ = f. Using a theorem of J. Saint Raymond we characterize functions equivalent to Borel measurable ones. This characterization answers a question asked by M. Morayne and C. Ryll-Nardzewski.

A function $f$ mapping the topological space $X$ to the space $Y$ is called a z-open function if for every cozeroset neighborhood $H$ of a zeroset $Z$ in $X$, the image $f\left(H\right)$ is a neighborhood of ${cl}_Y\left(f\left(Z\right)\right)$ in $Y$. We say $f$ has the z-separation property if whenever $U$, $V$ are cozerosets and $Z$ is a zeroset of $X$ such that $U\subseteq Z\subseteq V$, there is a zeroset $Z^{\text{'}}$ of $Y$ such that $f\left(U\right)\subseteq Z^{\text{'}}\subseteq f\left(V\right)$. A surjective function is z-open if and only if it maps cozerosets to cozerosets and has the z-separation property. We investigate z-open functions and other functions...

Some relationships between $1$-sequence-covering maps and weak-open maps or sequence-covering $s$-maps are discussed. These results are used to generalize a result from Lin S., Yan P., Sequence-covering maps of metric spaces, Topology Appl. 109 (2001), 301–314.