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 |