The disjoint arcs property for homogeneous curves

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. 0. 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 (DDP). The latter property turned out to be crucial in recognizing n-dimensional Menger space manifolds among LCn−1-spaces [2]. The DDP 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 DDP. 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. 1991 Mathematics Subject Classification: 54F50, 54F65.