International Journal of Theoretical Physics, Vol. 30, No. 7, 1991 Finitary Substitute for Continuous Topology Rafael D. Sorkin ~ Received January 4, 1991 Finite topological spaces are combinatorial structures that can serve as replace- ments for, or approximations to, bounded regions within continuous spaces such as manifolds. In this spirit, the present paper studies the approximation of general topological spaces by finite ones, or really by "finitary" ones in case the original space is unbounded. It describes how to associate a finitary space F with any locally finite covering of a Tl-space S; and it shows how F converges to S as the sets of the covering become finer and more numerous. It also explains the equivalent description of finite topological spaces in order-theoretic language, and presents in this connection some examples of posets F derived from simple spaces S. The finitary spaces considered here should not be confused with the so-called causal sets, but there may be a relation between the two notions in certain situations. 1. INTRODUCTION That matter on the smallest scales sheds its continuous nature is indicated by several features of present-day physics. In particular, the short-distance "cutoffs" required (apparently) by both quantum field theory (to "regularize" the functional integral) and "quantum gravity" (to render black hole entropy finite) seem ultimately foreign to the notion of differenti- able manifold embodied in classical general relativity. Their stubborn pres- ence suggests, rather, that there is a discrete substratum underlying space- time and accounting naturally for the appearance of a minimum length in the effective theories we now possess. Such an underlying discreteness, moreover, has often been looked to in the hope of finding explanations for such general features of nature as the existence of the spacetime metric, the presence of gauge and other fields interacting with this metric, the (3 + 1)-dimensionality of spacetime, the directionality of time, and the near vanishing of the cosmological constant. tlnstitute for Advanced Study, Princeton, New Jersey, 08540. Permanent address: Department of Physics, Syracuse University, Syracuse, New York 13244-1130. 923 0020-7748/91/0700-0923506.50/0 1991 Plenum Publishing Corporation