  It is pointed out that phase structures of gauge theories compactified on
non-simply connected spaces are not trivial. As a demonstration, an SU(2) gauge
model on $M^3\otimes S^1$ is studied and is shown to possess three phases:
Hosotani, Higgs and coexisting phases. The critical radius and the order of the
phase transitions are explicitly determined. A general discussion about phase
structures for small and large scales of compactified spaces is given. The
appearance of phase transitions suggests a GUT scenario in which the gauge
hierarchy problem is replaced by a dynamical problem of how to stabilize a
radius of a compactified space in close vicinity to a critical radius.
