Spacetimes as Topological Spaces, and the Need to Take Methods of General Topology More Seriously
Why is the manifold topology in a spacetime taken for granted? Why do we prefer to use Riemann open balls as basic-open sets, while there also exists a Lorentz metric? Which topology is a best candidate for a spacetime: a topology sufficient for the description of spacetime singularities or a topology which incorporates the causal structure? Or both? Is it more preferable to consider a topology with as many physical properties as possible, whose description might be complicated and counterintuitive, or a topology which can be described via a countable basis but misses some important information? These are just a few from the questions that we ask in this chapter, which serves as a critical review of the terrain and contains a survey with remarks, corrections and open questions.
KeywordsZeeman-Göbel topologies Topologising a spacetime Spacetime singularities Causal topologies Manifold topology
The co-author K.P. wishes to thank Robert Low for his remarks on the Interval Topology in , some of which we incorporate in the introductory section here, as well as Spiros Cotsakis for introducing him  and related literature. He also wishes to thank Nikolaos Kalogeropoulos for discussions on quantum theory of gravity and B.K. Papadopoulos for having taught him lattice topologies. Last, the authors would like to thank Ljubisa Kocinac and Hemen Dutta for their important remarks towards the improvement of the text.
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