The number of finite topologies

The logarithm (base 2) of the number of distinct topologies on a set of n elements is shown to be asymptotic to n2/4 as n goes to infinity. Let X be a set with n elements, and let T(X) be the set of all topologies that can be defined on X. Then we set T.= I T(X) I. The number T. has been determined for certain small values of n [4]. Tn has been estimated by several authors [2], [3], [8]. We present here an asymptotic estimate for the logarithm of Tn. We do this by considering an equivalent problem. Namely, if P(X) denotes the set of all partial orders that can be defined on a set X with n elements, and if we set Pn = I P(X) |, then we estimate Pn. The sizes of certain special subsets of P(X) have been determined, and thus provide lower bounds for Pn [6 ], [7 ]. We begin by presenting several enumeration problems which are equivalent to one another in a certain sense (see Lemma 1 below). Let Tn and Pn be defined as above. Similarly, for X a set with n elements, let To(X) be the set of all To-topologies that can be defined on X (i.e., if a, b are two elements of X, then there is some open set containing one but not both of them [5]), and let O(X) be the set of all preorders that can be defined on X (i.e., reflexive and transitive, but not necessarily antisymmetric). Then set Tn,o = I To(X) , and On = O(X) I. Let Tn, P', T' o, O' denote the numbers of isomorphism classes in T(X), P(X), To(X), O(X) respectively. From [2] we know that (1) ~~~~~Tn > 2n24 (Alternatively, we have Pn> 2n2 4 which we observe trivially after the introduction of "diagrams" below.) LEMMA 1. Tn = On, Tn,o=Pn, T' = 0', T' o=P. All eight of these quantities have logarithms which are asymptotically equal as n tends to infinity. Received by the editors, August 27, 1969. A MS Subject Classifications. Primary 0565.