Assuming countable choice, the cotangent bundle has a canonical smooth 2n-manifold structure

By [L1], a base chart $(U,x)$ induces a bijection ${\tilde{x}}^{\ast }:{\pi }^{-1}(U)\to x(U)\times {\mathbb{R}}^n$ using the coefficients in the basis $d{x}_p^1,\dots ,d{x}_p^n$. Declare the inverse images $({\tilde{x}}^{\ast }{)}^{-1}(O)$ of open sets $O$ to be basic open. The transition homeomorphisms in [L2] make these families agree on overlaps, so they define a topology in which every ${\tilde{x}}^{\ast }$ is a homeomorphism onto an open subset of ${\mathbb{R}}^{2n}$.

The projection $\pi :T^{\ast }M\to M$ is continuous because it is coordinate projection in every induced chart. The Hausdorff argument now separates covectors over distinct base points using [F2], and covectors over one point inside one Euclidean induced chart. Thus $T^{\ast }M$ is Hausdorff.

By [A1], [F2], and [L3], choose a countable subcover of $M$ by base-chart domains. Countable Euclidean bases in the corresponding induced charts pull back to a countable basis of $T^{\ast }M$, so $T^{\ast }M$ is second countable.

The transition maps are smooth with smooth inverses by [L2]. Together with steps 1.1-2.2, these charts define a canonical smooth $2n$-manifold structure on $T^{\ast }M$.