The ordinary definition of a tangent space uses the differentiable structure of differentiable manifolds and is hence not applicable to topological manifolds.
However for locally ringed spaces one can define the tangent space as the dual of the vector space obtained as a quotient by its maximal ideal, i.e. $\mathfrak{m}/\mathfrak{m}^2$.
Why is this latter construction not applicable to topological manifolds?
If you look at the ring of germs of functions of a topological manifold at a point (you might as well take the point to be the origin in the manifold $\Bbb R^n$), you'll get a local ring, but a very non-Noetherian one. If the ring is $A$ and the maximal ideal $m$, then $A/m\cong\Bbb R$, as it should, bur $m/m^2$ will be infinite-dimensional over $\Bbb R$, so it's difficult to use it as a cotangent space at that point.