  If $L$ is a finite lattice, we show that there is a natural topological
lattice structure on the geometric realization of its order complex $\Delta(L)$
(definition recalled). Lattice-theoretically, the resulting object is a
subdirect product of copies of $L$. We note properties of this construction and
of some variants thereof, and pose several questions. For $M_3$ the $5$-element
nondistributive modular lattice, $\Delta(M_3)$ is modular, but its underlying
topological space does not admit a structure of distributive lattice, answering
a question of Walter Taylor.
  We also describe a construction of "stitching together" a family of lattices
along a common chain, and note how $\Delta(M_3)$ can be obtained as a case of
this construction.
