Spacetime manifold

Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity. This physical theory models gravitation as the curvature of a four dimensional Lorentzian manifold (a spacetime) and the concepts of topology thus become important in analysing local as well as global aspects of spacetime. The study of spacetime topology is especially important in physical cosmology.

There are two main types of topology for a spacetime M.

As with any manifold, a spacetime possesses a natural manifold topology. Here the open sets are the image of open sets in ${\displaystyle \mathbb {R} ^{4}}$.

Definition: The topology ${\displaystyle \rho }$ in which a subset ${\displaystyle E\subset M}$ is open if for every timelike curve ${\displaystyle c}$ there is a set ${\displaystyle O}$ in the manifold topology such that ${\displaystyle E\cap c=O\cap c}$.

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