Definition: Cotangent Bundle
Let \(M\) be a differentiable manifold. The cotangent bundle of \(M\) is a set denoted by
\[T^{ * }M=\biguplus _{x\in M}T_{x}^{ * }M\,,\]
with the following properties:
 \(T^{ * }\) has an associated projection function
\[\pi \colon T^{ * }M\longrightarrow M,\,(x,u)\longmapsto x\,,\]
 \(T^{ * }\) has an associated topology, in which each subset \(W\subseteq T^{ * }M\) is open if and only if for each topological chart \(\alpha \colon U\longrightarrow V\,\) the set \((T^{ * }(\alpha ))^{1}\left(W\cap \pi ^{1}(U)\right)\) is open in \(V\times (\mathbb {R} ^{n})^{ * }\).
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References
Adapted from CC BYSA 3.0 Sources:
 Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of OsnabrÃ¼ck