Fixed point set of products and cones
John R. Martin, Lex G. Oversteegen, Edward D. Tymchatyn · Pacific Journal of Mathematics · 1982
A space X is said to have the complete invariance property (CIP) if every nonempty closed subset of X is the fixed point set of some self-map of X. Examples are given to show that for the class of locally connected continua, the operations of taking products, cones, and strong deformation retractions need not preserve CIP In fact, it is shown that the operations of taking products and cones do not preserve CIP for LC 00 continua.