  We consider an empty (4+1) dimensional Kaluza-Klein universe with a negative
cosmological constant and a Robertson-Walker type metric. It is shown that the
solutions to Einstein field equations have degenerate metric and exhibit
transitioins from a Euclidean to a Lorentzian domain. We then suggest a
mechanism, based on signature transition which leads to compactification of the
internal space in the Lorentzian region as $a \sim |\Lambda|^{1/2}$. With the
assumption of a very small value for the cosmological constant we find that the
size of the universe $R$ and the internal scale factor $a$ would be related
according to $Ra\sim 1$ in the Lorentzian region. The corresponding
Wheeler-DeWitt equation has exact solution in the mini-superspace giving rise
to a quantum state which peaks in the vicinity of the classical solutions
undergoing signature transition.
