Sr. No

Fine topology on space-time of GR

Diffeomorphism Group

Topological properties

1

HKM-path topology described by Hawking-King-McCarty (1976)

Conformal diffeomorphisms

Hausdorff, path connected and locally path connected, first countable, separable, but not normal or locally compact

2

Extended HKM-topology (Kim, 2006)

Conformal isomorphism group

Finer than Alexandrov topology

3

S-topology on Lorentz manifolds (Domiaty, 1985)

Conformal -diffeomorphisms

Hausdorff, first countable and separable, not regular and hence not metrizable, path connected and locally path connected

4

Zeeman -like fine topology in general relativity described by Göbel (1976)

Homeomorphism group with respect to Zeeman-like topology is the group of all homothetic transformations of V

Strongly causal space-times

5

Lindstrom (1978): Finest topology that induces the topology as a submanifold on time-like -curves and on space-like -curves

Group of Conformal -diffeomorphisms or group of all homothetic transformations of V

Space-time need not be strongly causal