Classifying spaces of degenerating mixed Hodge structures, I: Borel–Serre spaces

<!-- *** Custom HTML *** --> Let D be the classifying space of mixed Hodge structures with polarized graded quotients. We construct a real analytic manifold with corners DBS which contains D as a dense open subset. This is the mixed Hodge theoretic version of the Borel–Serre space in the case of pure weight constructed by Borel–Ji and Kato–Usui.