Maximal countably compact spaces and embeddings in MP-spaces

We study embeddings in maximal pseudocompact spaces together with maximal countable compactness in the class of Tychonoff spaces. It is proved that under MA +¬\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${+\neg}$$\end{document} CH any compact space of weight κ<c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa < \mathfrak{c}}$$\end{document} is a retract of a compact maximal pseudocompact space. If κ is strictly smaller than the first weakly inaccessible cardinal, then the Tychonoff cube [0, 1]κ is maximal countably compact. However, for a measurable cardinal κ, the Tychonoff cube of weight κ is not even embeddable in a maximal countably compact space. We also show that if X is a maximal countably compact space, then the functional tightness of X is countable. It is independent of ZFC whether every compact space of countable tightness must be maximal countably compact. On the other hand, any countably compact space X with the Mazur property (≡\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\equiv}$$\end{document} every real-valued sequentially continuous function on X is continuous) must be maximal countably compact. We prove that for any ω-monolithic compact space X, if Cp(X) has the Mazur property, then it is a Fréchet–Urysohn space.