Let $X$ be a paracompact smooth manifold. Let $\operatorname{Line}_{\mathbb C}(X)$ denote the set of isomorphism classes of smooth complex line bundles over $X$. Let $H^2(X;\mathbb Z)$ denote singular cohomology with integer coefficients, and for a smooth complex line bundle $L\to X$ let $c_1(L)\in H^2(X;\mathbb Z)$ denote its integral first Chern class. Then the map

\begin{align*} \operatorname{Line}_{\mathbb C}(X) &\longrightarrow H^2(X;\mathbb Z) \end{align*}

\begin{align*} [L] &\longmapsto c_1(L) \end{align*}

is a bijection.