Remarks on hereditarily indecomposable continua

Details Remarks on hereditarily indecomposable continua Remarks on hereditarily indecomposable continua

2000-10-25

arxiv.org/abs/math/0010234v1

Topology Proceedings, 25 (2000) Summer, 179-206

Mathematics

General Topology

Requires amsart.cls[2000/06/02]; contains one postscript figure

Scientific paper

We recall a characterization of hereditary indecomposability originally obtained by Krasinkiewicz and Minc, and show how it may be used to give unified constructions of various hereditarily indecomposable continua. In particular we answer a question asked by Mackowiak and Tymchatyn by showing that any continuum of arbitrary weight is a weakly confluent image of a hereditarily indecomposable continuum of the same weight. We present two methods of constructing these preimages: (a) by model-theoretic means, using the compactness and completeness theorems from first-order logic to derive these results for continua of uncountable weight from their metric counterparts; and (b) by constructing essential mappings from hereditarily indecomposable continua onto Tychonoff cubes. We finish by reviving an argument due to Kelley about hyperspaces of hereditarily indecomposable continua and show how it leads to a point-set argument that reduces Brouwer's Fixed-point theorem to its three-dimensional version.

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