Uncountable Family of 0-Rigid Continua that are Homeomorphic to Their Inverse Limits

Abstract

It is a well-known fact that there are continua X such that the inverse limit of any inverse sequence $$\{X,f_n\}$$ { X , f n } with surjective continuous bonding functions $$f_n$$ f n is homeomorphic to X. The pseudoarc or any Cook continuum are examples of such continua. Recently, a large family of continua X was constructed in such a way that X is $$\frac{1}{m}$$ 1 m -rigid and the inverse limit of any inverse sequence $$\{X,f_n\}$$ { X , f n } with surjective continuous bonding functions $$f_n$$ f n is homeomorphic to X by Banič and Kac. In this paper, we construct an uncountable family of pairwise non-homeomorphic continua X such that X is 0-rigid and prove that for any sequence $$(f_n)$$ ( f n ) of continuous surjections on X, the inverse limit $$\varprojlim \{X,f_n\}$$ lim ← { X , f n } is homeomorphic to X.