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.