Some Problems about Consonant and Co-Consonant Spaces

Abstract

In this paper, we first prove that the retract of a consonant space (or co-consonant space) is consonant (co-consonant). Simultaneously, we consider the co-consonance of two powerspace constructions and proved that (1) the co-consonance of the Smyth powerspace P_S(X) implies the co-consonance of X if X is strongly compact; (2) the co-consonance of X implies the co-consonance of the Smyth powerspace under some conditions; (3) if the lower powerspace P_H(X) is co-consonant, then X is co-consonant; (4) for a continuous poset P, the lower powerspace P_H(ΣP) is co-consonant.