I must apologize, I was a little bit excited when I began understanding some of this, I can not say I can compete with professionals, and words are still difficult concepts.
In (1) S.H. et al. it is discussed that in extremal cases of charged black holes in a euclidean space, there is a topological change as compared to the non extremal case. This is viewed as a discontinuity by (2) who attempts a similar calculation in AdS.
The notion of charge being associated with topology comes from some versions (3,4) of the calculation performed in (5). Where the "charges are Q1 and Q5 are generated by wrapping a number Q1 of D1-branes around the circle $x^5$ and a number Q5 of D5-branes around the five circles..." and N is the momentum around the circle $x^5$ (4). Since the entropy calculation relies on those charges, it should follow that the number of string states is also dependent on the topology.
So in (3, 4, 5), the calculation is done on a supersymmetric extremal black hole. In (1) the calculation is not on a supersymmetric extremal black hole, but there is difference in topology at the extremal limit, which is implied in (2) to be discontinuous. In both case the change in topology is related to their being two event horizons that do connect to each other, implying a toroidal shape. In (1) the second horizon extends to infinity, causing the change in topology, in (3), the two horizons are described as meeting at the same radius. This suggests some sort of degeneration of the torus, implying a change in topology.
My initial query is whether we should think of these changes in topology as discontinuous (assuming that what I described isn't horribly flawed), or possibly a smooth transitions due to quantum effects, such as uncertainty and tunneling, where the change only occur discontinuously at a microscopic level so that it appears smooth at a macroscopic level?
The second part of the question is that charges are described as being dependent on topology, and they are also related to symmetries. So it seems natural to think that symmetries are dependent on topology.
Although we live in a relativistic world, in our every day life, we still think in non-relativistic terms, and most of our theories have non-relativistic roots, so is it natural to think of there being some sort of low energy topological state (I don't know, something like Ball-3 x Real). That low energy state would apparently have certain symmetries that are dependent on that topology and would make it distinct from other possible topologies. Is that possible?
Further reference (pg 355 in Group Theory: An intuitive approach by R Mirman):