On the Number of Open Sets of Finite Topologies

Recent papers of Sharp [4] and Stephen [5] have shown that any finite topology with n points which is not discrete contains <(3/4)2” open sets, and that this inequality is best possible. We use the correspondence between finite TO-topologies and partial orders to find all non-homeomorphic topologies with n points and >(7/16)2” open sets. We determine which of these topologies are TO, and in the opposite direction we find finite T0 and non-T, topologies with a small number of open sets. The corresponding results for topologies on a finite set are also given. IfXis a finite topological space, thenXis determined by the minimal open sets U, containing each of its points x. X is a TO-space if and only if U, = U, implies x = y for all points x, y in X. If X is not To , the space 8 obtained by identifying all points x, y E X such that U, = U, , is a Tospace with the same lattice of open sets as X. Topological properties of the operation X-+x are discussed by McCord [3]. Thus for the present we restrict ourselves to To -spaces. If X is a finite TO-space, define x < y for x, y E X whenever U, C U, . This defines a partial ordering on X. Conversely, if P is any partially ordered set, we obtain a TO-topology on P by defining U, = {y/y 6 x} for x E P. The open sets of this topology are the ideals (also called semiideals) of P, i.e., subsets Q of P such that x E Q, y < x implies y E Q. Let P be a finite partially ordered set of order p, and define w(P) = j(P) 2-p, where j(P) is the number of ideals of P. If Q is another finite partially ordered set, let P + Q denote the disjoint union (direct sum) of P and Q. Thenj(P + Q) = j(P)j(Q) and w(P + Q) = w(P) w(Q). Let H, denote the partially ordered set consisting of p disjoint points, so w(H,) = 1. 74 ON THE NUMBER OF OPEN SETS OF FINITE TOPOLOGIES 75 THEOREM 1. If n 3 5, then up to homeomorphism there is one T,,-space with n points and 2” open sets, one with (3/4) 2” open sets, two with (5/8) 2” open sets, three with (9/16) 2n, two with (17/32) 2”, two with (l/2) 2”, two with (15/32) 2”, jive with (7/16) 2n, and for each m = 6, 7,..., n, two with (2”-l + 1) 2n-m. All other TO-spaces with n points have <(7/16) 2” open sets, giving a total of 2n + 8 with >(7/16) 2” open sets. PROOF. Consider the 18 partially ordered sets P, ,..., PIs of order 5 in Figure 1. Any partial order P weaker than some Pi (meaning Pi can be