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The disjoint arcs property for homogeneous curves

100%
Krupski P.
Fundamenta Mathematicae
|
1994-1995
|
tom 146
|
nr 2
159-169
EN
The local structure of homogeneous continua (curves) is studied. Components of open subsets of each homogeneous curve which is not a solenoid have the disjoint arcs property. If the curve is aposyndetic, then the components are nonplanar. A new characterization of solenoids is formulated: a continuum is a solenoid if and only if it is homogeneous, contains no terminal nontrivial subcontinua and small subcontinua are not ∞-ods.
2
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On the disjoint (0,N)-cells property for homogeneous ANR's

100%
Krupski P.
Colloquium Mathematicum
|
1993
|
tom 66
|
nr 1
77-84
EN
A metric space (X,ϱ) satisfies the disjoint (0,n)-cells property provided for each point x ∈ X, any map f of the n-cell $B^{n}$ into X and for each ε > 0 there exist a point y ∈ X and a map $g:B^{n} → X$ such that ϱ(x,y) < ε, $\widehat{ϱ}(f,g) < ε$ and $y ∉ g(B^{n})$. It is proved that each homogeneous locally compact ANR of dimension >2 has the disjoint (0,2)-cells property. If dimX = n > 0, X has the disjoint (0,n-1)-cells property and X is a locally compact $LC^{n-1}$-space then local homologies satisfy $H_{k}(X,X-x) = 0$ for k < n and H_{n}(X,X-x) ≠ 0.
3
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Hyperspaces of universal curves and 2-cells are true $F_{σδ}$-sets

100%
Krupski P.
Colloquium Mathematicum
|
2002
|
tom 91
|
nr 1
91-98
EN
It is shown that the following hyperspaces, endowed with the Hausdorff metric, are true absolute $F_{σδ}$-sets: (1) ℳ ²₁(X) of Sierpiński universal curves in a locally compact metric space X, provided ℳ ²₁(X) ≠ ∅ ; (2) ℳ ³₁(X) of Menger universal curves in a locally compact metric space X, provided ℳ ³₁(X) ≠ ∅ ; (3) 2-cells in the plane.
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