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Why is the manifold topology in a spacetime taken for granted? Why do weprefer to use Riemann open balls as basic-open sets, while there also exists aLorentz metric? Which topology is a best candidate for a spacetime; a topologysufficient for the description of spacetime singularities or a topology whichincorporates the causal structure? Or both? Is it more preferable to have atopology with as many physical properties as possible, whose description mightbe complicated and counterintuitive, or a topology which can be described via acountable basis but misses some important information? These are just a fewfrom the questions that we ask in this Chapter, which serves as a criticalreview of the terrain and contains a survey with remarks, corrections and openquestions.

Spacetimes as topological spaces, and the need to take methods of general topology more seriously