The disjoint arcs property for homogeneous curves

Abstract

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. Introduction.All spaces in the paper are metric separable and all maps are continuous. A space X is homogeneous if for each two points x, y ∈ X there exists a homeomorphism h : X → X such that h(x) = y. A curve is a one-dimensional continuum. A space X has the disjoint arcs property (DAP) if any two paths in X can be approximated, arbitrarily closely, by disjoint paths. More precisely, X has the DAP if for each ε > 0 and for any two maps f, g : I = [0, 1] → X there exist maps f , g : I → X such that f (I)∩g (I) = ∅ and (f, f ) < ε, (g, g ) < ε, where denotes the sup-norm metric induced by in X. If, in the definition, I is replaced by the ndimensional disk, one gets the disjoint n-disks property (DD n P ). The latter property turned out to be crucial in recognizing n-dimensional Menger space manifolds among LC n−1 -spaces [2]. The DD n P is a kind of a general position property, so it requires enough room-for manifolds this is guaranteed by the sufficiently high dimension, whereas Menger space manifolds have a special structure which is responsible for the DD n P.In [8] it was observed that all homogeneous locally compact locally connected spaces of dimension at least three have the DAP. Dimension two is not friendly to the DAP because no two-dimensional manifold has the property.

Local structure of homogeneous $ANR$-spaces

“…Hence, F has a minimal element B which is a carrier of j n A,B (γ). We also need the following version of Effros' theorem [11] for locally compact spaces (see [20,Proposition 1.4]): Theorem 2.2. Let X be a homogeneous space with a metric ρ, a ∈ X and ε > 0.…”

Homogeneous spaces not separated by arcs

“…Note that results of [3] imply that no region of homogeneous locally compact space X with dim X = n ≥ 1 can be separated by a closed subset of dimension ≤ n − 2. Thus Theorem 1.1 also holds in case dim X > 2.…”