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Serre-Swan

There is an equivalence of categories \begin{align*} {\mathsf{Top}}{ {\mathsf{Bun}}\qty{\operatorname{GL}_r} }_{/ {{\mathbf{C}}}} \qty{\scriptstyle \text{Hausdorff}} \rightleftharpoons\mathop{\mathrm{Proj}} {}_{C^0(X, {\mathbf{C}})}{\mathsf{Mod}} \qquad E\searrow X \mapsto {{\Gamma}\qty{E} } \end{align*}