We extend to Hausdorff continua the result of Camargo and Uzcátegui that says that if X is a metric continuum, Jones' set function T is continuous on singletons and T is idempotent on closed sets, ...then G={T({x})|x∈X} is a decomposition of X. We also present important implications of this result, a couple of them answer questions of the celebrated Houston Problem Book. We study Hausdorff continua with the property of Kelley and with the property of Kelley weakly. We establish a Hausdorff version of Jones' Aposyndetic Decomposition Theorem. To this end, we introduce the uniform property of Effros.
On manifolds with nonhomogeneous factors Cárdenas, Manuel; Lasheras, Francisco F.; Quintero, Antonio ...
Central European journal of mathematics,
06/2012, Letnik:
10, Številka:
3
Journal Article
Recenzirano
Odprti dostop
We present simple examples of finite-dimensional connected homogeneous spaces (they are actually topological manifolds) with nonhomogeneous and nonrigid factors. In particular, we give an elementary ...solution of an old problem in general topology concerning homogeneous spaces.
We study continuously irreducible continua and characterize them as those continua of type
λ for which the set function
T
is continuous. Using results by Mohler and Oversteegen, we present a new ...family of one-dimensional continua for which the set function
T
is continuous and no element of the family contains a pseudo-arc. We study the hyperspaces of these continua.
We answer in the negative the conjecture of Sam B. Nadler Jr and David P. Bellamy which says “
Let X be a homogeneous one-dimensional continuum. Then
T
is continuous for X”. We characterize the class ...of homogeneous continua for which
T
is continuous.
1 2 -Homogeneous continua with cut points Nadler, Sam B.; Pellicer-Covarrubias, Patricia; Puga, Isabel
Topology and its applications,
05/2007, Letnik:
154, Številka:
10
Journal Article
Recenzirano
Odprti dostop
A space is said to be
1
2
-
homogeneous provided that there are exactly two orbits for the action of the group of homeomorphisms of the space onto itself. It is shown that if
X is a
1
2
-homogeneous ...continuum with at least one cut point, then
X has either uncountably many cut points or only one cut point
c. In the former case,
X is
1
2
-homogeneous if and only if
X is an arc or
X is a compactification of the reals
R
1
whose remainder is the union of two disjoint, nondegenerate, homeomorphic homogeneous continua and the ends of
X are mutually homeomorphic and
1
3
-homogeneous. In the latter case, the closures of the components of
X
−
{
c
}
are mutually homeomorphic and 2-homogeneous at
c, and
ord
c
(
X
)
⩾
4
; furthermore, if
ord
c
(
X
)
⩽
ω
,
X is a locally connected bouquet of simple closed curves. Conversely, the two conditions about the components of
X
−
{
c
}
are shown to imply
X is
1
2
-homogeneous under an additional assumption, which is shown by examples to be both required and restrictive.
In this paper, we show that any finite product
N
n
of closed orientable surfaces of genus at least 2 satisfies the property that all proper, surjective mappings
p from an orientable (
n+2)-manifold
M ...to a 2-manifold
B for which each
p
−1 (
b) is homotopy equivalent to
N
n
necessarily are approximate fibrations.
For each
n greater than 4 and for each positive
k less than
n, examples of generalized
n-manifolds
X and cellular maps π from
R
n
onto
X are constructed having the following properties. The ...nonmanifold part of
X is homeomorphic to a
k-cell, and if
A is any closed subspace of
X of dimensions less than
k, then the decomposition of
R
n
induced over
A is shrinkable. In particular, the nonmanifold nature of
X is not detectable by examining closed subsets of
X of dimensions less than
k. These examples are produced by combining mixing techniques for producing generalized
n-manifolds whose nonmanifold part is a Cantor set, with decompositions arising from special functions from the Cantor set onto a
k-cell
Studied here are the closed, connected manifolds N" with the property that all proper, surjective mappings p from an (n+2)-manifold M to a finite-dimensional ANR
B for which each
p
−1(
b) is ...homeomorphic to N
n necessarily are approximate fibrations. All simply connected
N
tn
are known to possess this property. This paper demonstrates that projective n-spaces
P
n
(
n > 1) and all closed surfaces
N
2 with nonzero Euler characteristic possess it as well.
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