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The origin of spacetime topology and generalizations of quantum field theory

The research effort reported in this paper is directed, in a broad sense, towards understanding the small-scale structure of spacetime. The fundamental question that guides our discussion is ``what is the physical content of spacetime topology?" In classical physics, if spacetime, $(X, τ)$, has sufficiently regular topology, and if sufficiently many fields exist to allow us to observe all continuous functions on $X$, then this collection of continuous functions uniquely determines both the set of points $X$ and the topology $τ$ on it. To explore the small-scale structure of spacetime, we are led to consider the physical fields (the observables) not as classical (continuous functions) but as quantum operators, and the fundamental observable as not the collection of all continuous functions but the local algebra of quantum field operators. In pursuing our approach further, we develop a number of generalizations of quantum field theory through which it becomes possible to talk about quantum fields defined on arbitrary topological spaces. Our ultimate generalization dispenses with the fixed background topological space altogether and proposes that the fundamental observable should be taken as a lattice (or more specifically a ``frame," in the sense of set theory) of closed subalgebras of an abstract $C^{\ast}$ algebra. Our discussion concludes with the definition and some elementary