Spacetime topology

Types of topology

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

Manifold topology

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

Path or Zeeman topology

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 ${\displaystyle E\cap c=O\cap c}$.

It is the finest topology which induces the same topology as ${\displaystyle M}$ does on timelike curves.

Properties

Strictly finer than the manifold topology. It is therefore Hausdorff, separable but not locally compact.

A base for the topology is sets of the form ${\displaystyle Y^{+}(p,U)\cup Y^{-}(p,U)\cup p}$ for some point ${\displaystyle p\in M}$ and some convex normal neighbourhood ${\displaystyle U\subset M}$.

(${\displaystyle Y^{\pm }}$ denote the chronological past and future).

Alexandrov topology

The Alexandrov topology on spacetime, is the coarsest topology such that both ${\displaystyle Y^{+}(E)}$ and ${\displaystyle Y^{-}(E)}$ are open for all subsets ${\displaystyle E\subset M}$.

Here the base of open sets for the topology are sets of the form ${\displaystyle Y^{+}(x)\cap Y^{-}(y)}$ for some points ${\displaystyle x,y\in M}$.

This topology coincides with the manifold topology if and only if the manifold is strongly causal but it is coarser in general.

Note that in mathematics, an Alexandrov topology on a partial order is usually taken to be the coarsest topology in which only the upper sets ${\displaystyle Y^{+}(E)}$ are required to be open. This topology goes back to Pavel Alexandrov.

Nowadays, the correct mathematical term for the Alexandrov topology on spacetime (which goes back to Alexandr D. Alexandrov) would be the interval topology, but when Kronheimer and Penrose introduced the term this difference in nomenclature was not as clear, and in physics the term Alexandrov topology remains in use.