In his paper "Categories and cohomology theories" Graeme Segal considers the category of finite length chain complexes of finite dimensional vector spaces: Let $n = (n_i)_{i \in \mathbb{Z}}$ be a sequence of positive integers almost all zero. Then he claims that the space $K_n$ of chain complexes $E$ with $E^i = \mathbb{R}^{n_i}$ is a real algebraic variety. I am no algebraic geometer, but I suspect this to be similar to flag varieties somehow. So my first question is:

What is the topology on $K_n$?

As morphisms in the above mentioned category he takes *chain homotopy equivalences* between these complexes and claims that these also form a topological space. So:

What is the topology on Mor($E,F$) for two chain complexes $E$ and $F$ as described above?