Stephen Hawking, among others, has proposed that the topological stability of a property of space-time is a necessary condition for it to be physically significant. What counts as stable, however, depends crucially on the choice of topology. Some physicists have thus suggested that one should find a canonical topology, a single 'right' topology for every inquiry. While certain such choices might be initially motivated, some little-discussed examples of Robert Geroch and some propositions of my own show that the main candidates - and each possible choice, to some extent - faces the horns of a no-go result. I suggest that instead of trying to decide what the 'right' topology is for all problems, one should let the details of particular types of problems guide the choice of an appropriate topology.
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