On Extremal Spaces
by
Vinod Kumar
Kurukshetra University, Kurukshetra - 136119 INDIA

Hochster [1] has obtained, using Zorn's lemma, a characterisation of an extremal space X in terms of retraction of the power set of X upon all clopen sets of X. We prove a variation of this result without using Zorn's lemma or its equivalent or weaker forms. We introduce idempotent operators \gamma( cl o int ) and \alpha( int o cl ). \gamma ( \alpha) is shown to be distributive over union (intersection) of two subsets of X if one of them is either open or closed. X is extremal if and only if \gamma ( or equivalently \alpha ) is distributive over intersection (union) of any two subsets of X.

[1] Hochster M., extremal spaces, Pac. J. Math. 32, (1970)767-779.

Date received: June 17, 1999

Copyright © 1999 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 # cacl-84.