  The formalism of Greensite for treating the spacetime signature as a
dynamical degree of freedom induced by quantum fields is considered for
spacetimes with nontrivial topology of the kind ${\bf R}^{D-1} \times {\bf
T}^1$, for varying $D$. It is shown that a dynamical origin for the Lorentzian
signature is possible in the five-dimensional space ${\bf R}^4 \times {\bf
T}^1$ with small torus radius (periodic boundary conditions), as well as in
four-dimensional space with trivial topology. Hence, the possibility exists
that the early universe might have been of the Kaluza-Klein type, \ie
multidimensional and of Lorentzian signature.
