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While looking at Prof. Brian Greene's research work I came across the statement regarding the difference between quantum geometry and the geometry underlying general relativity, "...topology change (the "tearing" of space) is a sensible feature of quantum geometry even though, from a classical perspective, it involves singularities."

As someone who has recently started studying topology, is my guess that the reason singularities not allowed in classical geometry is because isolated points are not open sets in the open ball Euclidean topology anywhere close to correct ?

Or is the Euclidean topology not applicable for general relativity in the first place ?

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