Classical topology and quantum states

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomo rphic. The sets of all their self-adjoint operators are also therefore unit arily equivalent. Thus if all self-adjoint operators can be observed, and i f there is no further major axiom in quantum physics than those formulated for example in Dirac's 'quantum mechanics', then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving observables with smooth time evolution: th ey contain commutative subalgebras from which the spatial slice of spacetim e with its topology (and with further refinements of the axiom, its C-K- an d C-infinity-structures) can be reconstructed using Gel'fand-Naimark theory and its extensions. Classical topology is an attribute of only certain qua ntum observables for these axioms, the spatial slice emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After for mulating these axioms, we apply them to show the possibility of topology ch ange and to discuss quantized fuzzy topologies. Fundamental issues concerni ng the role of time in quantum physics are also addressed.