Planar Completely Regular Continua with Structure
by
S. S. Zafiridou
Patras, Greece

Planar Completely Regular Continua with Structure

S.S. Zafiridou

(Patras, Greece)

Abstract

A metric continuum K is called completely regular if each nondegenerate subcontinuum of K has nonvoid interior. For every completely regular continuum K there exist a homeomorphism f of the Cantor set C into K and homeomorphisms f_k:[0, 1] --> K, k=1, 2, ..., such that: 1)K=f(C) \cup ( \cup _k=1^\inftyf_k([0, 1])), 2)f_k((0, 1)) is open in K and f_k([0, 1]) \cap f(C)=f_k({ 0, 1} ), 3)f_k([0, 1]) \cap f_m([0, 1])=\emptyset for every m =/= n, and 4)limdiam f_k([0, 1])=0 (see [1]).

The triple (K, F, A), where F=f(C) and A={f_k([0, 1]):k >= 1} is called completely regular continuum with structure. A completely regular continuum with structure ([K\tilde], [F\tilde], [A\tilde]) is said to be universal for a family F of completely regular continua with structure if ([K\tilde], [F\tilde], [A\tilde]) in F and for every (K, F, A) in F there exists a homeomorphism h:K --> [K\tilde] such that h(F) subset or equal [F\tilde] and h(a) in [A\tilde] for every a in A (see [1]).

It is known that:

(1) In the class of all completely regular continua there exists a universal element (see [1]).

(2) In the class of all completely regular continua with structure there is no universal element (see [2]).

We shall prove that in the class of all planar completely regular continua with structure there is no universal element.

References.

[1] S.D. Iliadis, Universal continuum for the class of completely regular continua, Bull. de L'academie Polonaise Des Science, Vol. XXVIII, No. 11-12, 1980, 603-607.

[2] S.D. Iliadis, Rim-finite spaces and the property of universality, Houston Journal of Math., Vol. 12, No. 1, 1986, 55-78.

Date received: June 15, 1998

Copyright © 1998 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 # cabc-30.