Can the Cook curve construction be modified in some special way? by Jun Terasawa

Topology Atlas Document # zaaa-41.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 4 (1996), page 67.

Can the Cook curve construction be modified in some special way?

(Department of Mathematics, The National Defense Academy, Yokosuka, Japan)

The Cook Curve is a hereditarily indecomposable one-dimensional continuum in which any continuous map between two non-degenerate subcontinua is the identity. (see H.Cook, "Continua which admit only the identity mapping onto non-degenerate subcontinua", Fund.Math. 60(1967) pp. 241-249) It is constructed using inverse limits and multivalued functions and built up by adding circles at each stage. The question is whether it is possible to construct similar spaces in the fashion of Cantor's middle-thirds set, by removing subsets step by step from, say, the unit cube?

This question is closely related to my article "Rigid spaces" in Topology Atlas Invited Contributions, vol. 1, issue 4 (1996), pages 58-59.

Received by the editors: January 8, 1996. Revised: January 16, 1996.