Quantum topology and quantisation on the lattice of topologies
Christopher John Isham · Classical and Quantum Gravity · 1989
The concept of 'quantum topology' is studied via a quantisation of the set tau (X) of all topologies on a given set X. A natural lattice structure exists on this set induced by the idea of one topology having more, or less, open sets than other. This is used to provide a basic set of functions on tau (X) which generate a commutative algebra (the v operation on the lattice) whose spectral theory forms the basis for a general quantisation. It is shown that the analogue of a 'distributional' topology is an ideal in the lattice tau (X) and the spectral theory is used to place a natural topology on the set of all such ideals. The next step is to discuss the existence of variables conjugate to the basic functions on tau (X).