What is a non-metrizable analog of metrizable compacta? (Part I)

Two classes of compacta were introduced: the class of metrcompacta and more wide class of weak metrcompacta. Both classes are countably productive. The class of weak metrcompacta is a strict subclass of uniform Eberlein compacta. For any cardinal number τ, there exists a rather simple metrcompactum that is a topologically universal element in the class of all weak metrcompacta (and metrcompacta) of weight τ. Every weak metrcompactum (in particular, every metrcompactum) has a 0-dimensional map onto a metrizable compactum and so the dimensions dim, ind, Ind and Δ coincide for all weak metrcompacta (and metrcompacta). Every metrizable space X has a compactification cX that is a metrcompactum with .