Free topological semilattices homeomorphic to R∞ or Q∞

Let F∞(X) be the free topological semilattice over a kω -space X (i.e., the direct limit of a tower of compacta). It is proved that F∞(X) is homeomorphic to R∞=lim→Rn if and only if X is connected, X has no isolated points and every compactum in X is contained in a finite-dimensional locally connected compact metrizable subset of X . It is also shown that F∞(X) is homeomorphic to Q∞=lim→Qn if X is connected and every compact subset of X lies in a Q -manifold M⊂X , where Q=[−1,1]ω is the Hilbert cube. In the above, in case X is not connected, F∞(X) is locally homeomorphic to R∞ or Q∞ .