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It is known that no dendrite (Gottschalk 1947) and no hereditarily indecomposable tree-like continuum (J. Heath 1991) can be the image of a continuum under an exactly 2-to-1 (continuous) map. This paper enlarges the class of tree-like continua satisfying this property, namely to include those tree-like continua whose nondegenerate proper subcontinua are arcs. This includes all Knaster continua and Ingram continua. The conjecture that all tree-like continua have this property, stated by S. Nadler...
1991 AMS Math. Subj. Class.:Primary 54C10; Secondary 54F65We provide both a spectral and an internal characterizations of arbitrary !-favorable spaces with respect to co-zero sets. As a corollary we establish that any product of compact !-favorable spaces with respect to co-zero sets is also !-favorable with respect to co-zero sets. We also prove that every C* -embedded !-favorable with respect to co-zero sets subspace of an extremally disconnected space is extremally disconnected.
The following theorem is proved. Let f: X → Y be a finite-to-one map such that the restriction is an inductively perfect map for every countable compact set S ⊂ Y. Then Y is a countable union of closed subsets such that every restriction 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 -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 mapping the topological space to the space is called a z-open function if for every cozeroset neighborhood of a zeroset in , the image is a neighborhood of in . We say has the z-separation property if whenever , are cozerosets and is a zeroset of such that , there is a zeroset of such that . 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...
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