On topologies of triangulated infinite-dimensional manifolds (Q1107177)

Let R n be considered as the subset of the countable infinite product \(R^{\omega}\) of the real line R. The set \(\cup_{n\in N}R\) n admits two different topologies. One is the weak topology with respect to the tower \(\{\) R \(n\}_{n\in N}\) and the space with this topology is denoted by \(R^{\infty}\). Another is the relative topology inherited from the product topology of \(R^{\omega}\) and the space with this topology is denoted by \(\sigma\). A separable topological manifold modeled on these spaces is called an \(R^{\infty}\)-manifold and \(\sigma\)-manifold, respectively. It is known that every \(R^{\infty}\)-manifold is homeomorphic to a simplicial complex with a weak topology and every \(\sigma\)-manifold is homeomorphic to a simplicial complex with the metric topology. Let K be a simplicial complex and \(| K| =\cup K\) be the realization of K. By \(| K|_ w\) and \(| K|_ m\) is denoted the spaces \(| K|\) with the weak topology and the metric topology, respectively. The main result of the paper is: for a simplicial complex K \(| K|_ m\) is a \(\sigma\)-manifold if \(| K|_ w\) is an \(R^{\infty}\)-manifold.