Wikipedia says that the tangent spaces of a connected manifold all have the same dimension, equal to that of the manifold.
Well, is there an example of a simple disconnected manifold that doesn't have this property?
All I can think of is taking $[0,1]$ and $[2,3]$ as a subspace of $\mathbb{R}$. Well, first of all, does the tangent space even exist at $0,1,2,3$? I can see that the set of "virtual velocities" are only in one direction at these points. Hence, this does not constitute a linear space. Is it a tangent space then?
If it is a tangent space, its dimension would still be 1, as the closed half-line of $\mathbb{R}$, so I am failing to find a good counterexample.
Just to close the question: You can take a disjoint union of two manifolds of different dimensions. Some people see this space as a manifold.