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In mathematics, the circle group, denoted by T {\displaystyle \mathbb {T} } or S {\displaystyle \mathbb {S} } , is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers
The circle group plays a central role in Pontryagin duality, and in the theory of Lie groups.
The notation T {\displaystyle \mathbb {T} } for the circle group stems from the fact that, with the standard topology , the circle group is a 1-torus. More generally T n {\displaystyle \mathbb {T} ^{n}} is geometrically an n {\displaystyle n} -torus.