A universal model infinite-dimensional space

Given an ordinal αα and a pointed topological space XX, we endow X<α=∪{Xβ:β<α}X<α=∪{Xβ:β<α} with the strongest topology that coincides with the product topology on every subset XβXβ of X<αX<α, β<αβ<α. It turns out that many important model spaces of infinite-dimensional topology (including the topology of non-metrizable manifolds) can be obtained as spaces of the form X<αX<α for X=I,RX=I,R. This paper deals with some topological properties of spaces X<αX<α. Some new classification and characterization theorems are proved for these spaces.