  We study the classical and quantum Euclidean wormholes for an empty (4+1)
dimensional Kaluza-Klein universe with a positive cosmological constant and a
spatially flat Robertson-Walker type metric. It is shown that classical
wormholes do not exist neither for the spacetime sector nor the extra
dimensional sector of this model, but two spectrums of quantum wormholes as the
solutions of Wheeler-DeWitt equation exist which are consistent with the
Hawking-Page conjecture about the necessary boundary conditions. In the
spectrum where the external scale factor $R$ shapes the quantum wormholes the
internal scale factor $a$ may play the role of effective matter source, and in
the spectrum where the internal scale factor $a$ shapes the quantum wormholes
the external scale factor $R$ may play the role of effective matter source.
