Trousers
The trousers topology is the standard example of a spacetime with topology change.
1. History
2. Topology
The trousers spacetime has the topology of the connected sum of a cylinder and the plane, $(S \times \mathbb{R}) \# \mathbb{R}^2$, or equivalently, a sphere with three disks removed, $S^2 \setminus \{D_1^2, D_2^2, D_3^2\}$. It is a Lorentz cobordism between two circles and one circle, that is, a Lorentz cobordism with boundaries $S^1 \sqcup S^1$ and $S^1$.
3. Metrics and coordinates
While many different metrics are possible on it, the most commonly used one is simply a Minkowski metric
$$ds^2 = -dt^2 + dx^2$$