The dimensional structure of hereditarily indecomposable continua
by
Roman Pol
Warsaw University, Poland
Coauthors: Mirosława Reńska (Warsaw University)

A continuum X is hereditarily indecomposable, abbreviated HI, if for any pair of intersecting continua A, B in X, either A subset B or B subset A. Bing constructed in 1951 HI continua of all finite dimensions and pointed out that in any r-dimensional HI continuum X there are points not belonging to any non-trivial continuum in X with dimension less than r.

Theorem. Let X be an n-dimensional hereditarily indecomposable continuum and let B_r be the set of all points in X that belong to some r-dimensional continuum but avoid any non-trivial continuum of dimension less than r. Then dim(B_n \cup (B_r\N)) = n - (r-1) for any N with dimN <= 0.

One can derive from this theorem that dim(B_n \C) = 1 for any \sigma-compact (n-2)-dimensional set C in X and that dimB_r = n -(r-1).

The sets B_r, for r >= 2, are never G_\delta\sigma-sets, being always G_{\delta\sigma\delta}-sets in X.

For any r with 1 < r < n there is a point b in B_r and \epsilon > 0 such that no non-trivial connected set in X \B₁ of diameter less than \epsilon touches b. This yields some refinements of the remarkable fact established by Bing that no n-dimensional HI continuum X with n >= 2 is homogeneus.

Date received: June 22, 2000

Copyright © 2000 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 # caeu-42.