The Disjoint Arcs Property and Homogeneity
by
Pawel Krupski
University of Wroclaw

A space X is homogeneous if for each two points x, y in X there exists a homeomorphism h : X --> X such that h(x) = y. A metric space X has the disjoint arcs property (DAP) if any two paths in X can be approximated, arbitrarily closely, by disjoint paths.

Developing ideas from [1] and [2] one can get the following improvements of some results of those papers.

Theorem 1. If X is a homogeneous metric continuum, then one of the following cases holds:

• X is a solenoid;
• X is a closed 2-manifold;
• X is locally connected with the DAP;
• there exists an \alpha > 0 such that each open nonempty subset of X of diameter < \alpha consists of uncountably many nowhere dense components which are locally connected and have the DAP;
• each component C of an arbitrary open nonempty subset of X has an open (in C) cover by subsets of C consisting of uncountably many components nowhere dense in C.

Theorem 2. If X is a homogeneous metric continuum which is not a solenoid nor a 2-manifold, then each component of an arbitrary open subset of X has the DAP.

As a corollary we obtain an interesting Characterization of closed 2-manifolds. For any metric continuum X, if n <= 2, then X is a closed 2-manifold if and only if X is homogeneous and does not have the DAP.

1 P. Krupski The disjoint arcs property for homogeneous curves Fund. Math 1995 146 159-169
2 P. Krupski and H. Patkowska Menger curves in Peano continua Colloq. Math. 1996 79-86 70

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-58.