The Up Topology for Mirrored Topological Posets

Abstract

For a discrete poset $\mathcal X$, McCord proved that the natural map $|{\mathcal X}|\to {\mathcal X}$, from the order complex to the poset with the Up topology, is a weak homotopy equivalence. Much later, \u{Z}ivaljevi\'{c} defined the notion of order complex for a topological poset. For a large class of such topological posets we prove the analog of McCord's theorem, namely that the natural map from the order complex to the topological poset with the Up topology is a weak homotopy equivalence. An example is the Grassmann poset of proper non-zero linear subspaces of $\R^{n+1}$. Here, Vassiliev had computed the homotopy type of the order complex. Our theorem allows us to transfer that information (up to weak homotopy type) to the Grassmann poset itself with the Up topology.