Displaying similar documents to “Property Z for function-graphs and finite-dimensional sets in I ∞ and s ”

A note on splittable spaces

Vladimir Vladimirovich Tkachuk (1992)

Commentationes Mathematicae Universitatis Carolinae

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A space X is splittable over a space Y (or splits over Y ) if for every A ⊂ X there exists a continuous map f : X → Y with f - 1 f A = A . We prove that any n -dimensional polyhedron splits over 𝐑 2 n but not necessarily over 𝐑 2 n - 2 . It is established that if a metrizable compact X splits over 𝐑 n , then dim X ≤ n . An example of n -dimensional compact space which does not split over 𝐑 2 n is given.

More on tie-points and homeomorphism in ℕ*

Alan Dow, Saharon Shelah (2009)

Fundamenta Mathematicae

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A point x is a (bow) tie-point of a space X if X∖x can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as X = A ⋈ x B where A, B are the closed sets which have a unique common accumulation point x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) of precisely 2-to-1 maps on ℕ*. In these cases the tie-points have been the unique...

A Bing-Borsuk retract which contains a 2-dimensional absolute retract

Steve Armentrout

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Contents1. Introduction .................................................................................................. 52. Antoine’s necklaces.................................................................................... 63. Wreaths......................................................................................................... 74. Construction of discs.................................................................................. 75. Construction of Bing-Borsuk...

Filling boxes densely and disjointly

J. Schröder (2003)

Commentationes Mathematicae Universitatis Carolinae

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We effectively construct in the Hilbert cube ℍ = [ 0 , 1 ] ω two sets V , W ⊂ ℍ with the following properties: (a) V ∩ W = ∅ , (b) V ∪ W is discrete-dense, i.e. dense in [ 0 , 1 ] D ω , where [ 0 , 1 ] D denotes the unit interval equipped with the discrete topology, (c) V , W are open in ℍ . In fact, V = ⋃ ℕ V i , W = ⋃ ℕ W i , where V i = ⋃ 0 2 i - 1 - 1 V i j , W i = ⋃ 0 2 i - 1 - 1 W i j . V i j , W i j are basic open sets and ( 0 , 0 , 0 , ... ) ∈ V i j , ( 1 , 1 , 1 , ... ) ∈ W i j , (d) V i ∪ W i , i ∈ ℕ is point symmetric about ( 1 / 2 , 1 / 2 , 1 / 2 , ... ) . Instead of [ 0 , 1 ] we could have taken any T 4 -space or a digital interval, where the resolution (number of points) increases with i .

Reconstruction of manifolds and subsets of normed spaces from subgroups of their homeomorphism groups

Matatyahu Rubin, Yosef Yomdin

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This work concerns topological spaces of the following types: open subsets of normed vector spaces, manifolds over normed vector spaces, the closures of open subsets of normed vector spaces and some other types of topological spaces related to the above. We show that such spaces are determined by various subgroups of their auto-homeomorphism groups. Theorems 1-3 below are typical examples of the results obtained in this work. Theorem 1. For a metric space X let UC(X) denote the group...

Decomposition Results for Functions with Bounded Variation

Gianni Dal Maso, Rodica Toader (2008)

Bollettino dell'Unione Matematica Italiana

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Some decomposition results for functions with bounded variation are obtained by using Gagliardo's Theorem on the surjectivity of the trace operator from W 1 ; 1 ⁢ ( Ω ) into L 1 ⁢ ( ∂ ⁡ Ω ) . More precisely, we prove that every BV function can be written as the sum of a BV function without jumps and a BV function without Cantor part. Alternatively, it can be written as the sum of a BV function without jumps and a purely singular BV function (i.e., a function whose gradient is singular with respect to the Lebesgue...

On isometrical extension properties of function spaces

Hisao Kato (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces C ( Q ) and C ( Δ ) , where Q and Δ denote the Hilbert cube [ 0 , 1 ] ∞ and a Cantor set, respectively.

Maps with dimensionally restricted fibers

Vesko Valov (2011)

Colloquium Mathematicae

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We prove that if f: X → Y is a closed surjective map between metric spaces such that every fiber f - 1 ( y ) belongs to a class S of spaces, then there exists an F σ -set A ⊂ X such that A ∈ S and d i m f - 1 ( y ) ∖ A = 0 for all y ∈ Y. Here, S can be one of the following classes: (i) M: e-dim M ≤ K for some CW-complex K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = M: dim M ≤ n, then dim f ∆ g ≤ 0 for almost all g ∈ C ( X , n + 1 ) .

On Surjective Bing Maps

Hisao Kato, Eiichi Matsuhashi (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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In [7], M. Levin proved that the set of all Bing maps of a compact metric space to the unit interval is a dense G δ -subset of the space of all maps. In [6], J. Krasinkiewicz independently proved that the set of all Bing maps of a compact metric space to an n-dimensional manifold (n ≥ 1) is a dense G δ -subset of the space of maps. In [9], J. Song and E. D. Tymchatyn, solving some problems of J. Krasinkiewicz ([6]), proved that the set of all Bing maps of a compact metric space to a nondegenerate...