To my lay-person mind, a chart is a one-to-one function that maps an area on a manifold to a euclidean space of equal dimension.
Then I understand a tangent space to be the space of vectors that are parallel to the manifold at a specific point.
So does one use the appropriate charts to map points to the tangent space?
These seem like similar things to me (even though I am fairly certain they are not). Is the difference that any point on the manifold can be mapped to that specific tangent space whereas the chart only covers a subset of points in the manifold?
Or do I have two completely different ideas conflated?
Is the mapping of points from the manifold even related to the manifolds charts?