A universal model infinite-dimensional space

Given an ordinal $\alpha$ and a pointed topological space $X$, we endow $X^{<\alpha }=\cup \{X^{\beta }:\beta <\alpha \}$ with the strongest topology that coincides with the product topology on every subset $X^{\beta }$ of $X^{<\alpha }$, $\beta <\alpha$. 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 $X^{<\alpha }$ for $X=I,R$. This paper deals with some topological properties of spaces $X^{<\alpha }$. Some new classification and characterization theorems are proved for these spaces.