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Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.

https://en.wikipedia.org/wiki/Subspace_topology

My question is regarding how subsets of topological spaces can be equipped with a different topology? In particular, how can a subspace of spacetime, like the interior of a black hole be treated with a different topology? Specifically, can the interior of a black hole be treated as a non-metrizable or non-Hausdorff space, and how could that affect bulk-edge correspondence? Is that inconsistent with other theorems or just too weird to ask from black holes? Does it contradict with black holes having a finite entropy, and introduce weird non-locality? How would a non-metrizable or non-Hausdorff black hole be described in some AdS-CFT, what kinds of geometry can be done on such analysis?

What effect could such treatment of black holes have on something like holographic superconductors? https://arxiv.org/abs/0810.1563

Lots of other ideas and discussions could follow if that is doable.

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Mainstream GR says that the interior of a black hole has a Lorentzian metric just like the exterior does. Other than a singularity inside, there is nothing particularly weird (such as non-metrizability) going on. You can, for example, predict your perfectly smooth trajectory inside until you reach the singularity. The global topology may be exotic (wormholes, anyone?) but the local topology is perfectly normal inside the horizon but outside the singularity.

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  • $\begingroup$ Does it make sense to do an exotic topology just on the singularity assuming either the singularity to be a single point entity, or perhaps better without that assumption? Does it help in describing exotic quantum states inside a black hole? I wonder if that could allow some kind of non-locality confined to inside the event horizon, that could be interesting in some way. $\endgroup$
    – Kaushal Timilsina
    Commented Jan 11, 2020 at 15:53
  • $\begingroup$ Sorry, I don’t have an answer to those questions. Perhaps someone else can comment on them, or you can post a new question. $\endgroup$
    – G. Smith
    Commented Jan 12, 2020 at 0:45
  • $\begingroup$ Thanks for your answer! I expected to initiate a discussion about this topic that others may find useful and interesting, and perhaps get diverse ideas than a settled answer. $\endgroup$
    – Kaushal Timilsina
    Commented Jan 12, 2020 at 5:26
  • $\begingroup$ @safesphere I will keep the site policies in mind. A.V.S. gives a great answer to the event horizon connecting the outside and the inside spacetime. A.V.S also points out to something I am interested in- a manifold that's a surface on its own: "One could even go further, and identify the mirror points on two sheets of the Einstein-Rosen solution and obtain a spacetime without any interior: the spacetime simply ends at the null boundary." I asked a question of similar interest math.stackexchange.com/questions/3505264/… Thanks! $\endgroup$
    – Kaushal Timilsina
    Commented Jan 12, 2020 at 6:54

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