Compact spaces generated by retractions

Abstract

We study compact spaces which are obtained from metric compacta by iterating the operation of inverse limit of continuous sequences of retractions. This class, denoted by R , has been introduced in [M. Burke, W. Kubiś, S. Todorčević, Kadec norms on spaces of continuous functions, http://arxiv.org/abs/math.FA/0312013]. Allowing continuous images in the definition of class R , one obtains a strictly larger class, which we denote by R C . We show that every space in class R C is either Corson compact or else contains a copy of the ordinal segment ω 1 + 1 . This improves a result of Kalenda from [O. Kalenda, Embedding of the ordinal segment [ 0 , ω 1 ] into continuous images of Valdivia compacta, Comment. Math. Univ. Carolin. 40 (4) (1999) 777–783], where the same was proved for the class of continuous images of Valdivia compacta. We prove that spaces in class R do not contain cutting P-points (see the definition below), which provides a tool for finding spaces in R C ∖ R . Finally, we study linearly ordered spaces in class R C . We prove that scattered linearly ordered compacta belong to R C and we characterize those ones which belong to R . We show that there are only 5 types (up to order isomorphism) of connected linearly ordered spaces in class R and all of them are Valdivia compact. Finally, we find a universal pre-image for the class of all linearly ordered Valdivia compacta.