We consider an SU(2) gauge theory in six-dimensions. We show that there exists non-trivial structure of gauge vacua in compactified background M4xS2. In this circumstances, there can exist a cosmic string, whose solution is recently constructed by the present authors. We consider the stage of the formation of the strings, i.e., the phase transition of the gauge configuration on the sphere in the 'hot' early universe. We find that if SU(2)-doublet fermions are the dominant component of the

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... ion at high temperatures, the phase transition successively occurs after the spontaneous-compactification transition. Otherwise, non-trivial vacuum with 'monopole' configuration of the SU(2) gauge field never appears globally and strings must be formed at the same time as the extra space is compactified.