Topology/Tangent Spaces

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Briefly, a tangent is a derivative of a curve. Translated into topology this means that you can effectively remove one dimension from a picture. Typically, when working with space time we can can perform one of two operations: Either remove time as a way to see frozen 3D images or we can remove one spatial dimension and therefore represent space time as a curved surface (the net like drawings so typically used to represent topological surfaces in pictures or such phenomena as black holes which are often drawn as shrinking cones).

Tangent spaces are therefore only a representation of what we understand to be one dimension simpler than the problems of topology. It is a useful tool for visualising space time arguments and positions.