On the lattice of topologies

Juris Hartmanis
1958 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
In many cases Lattice Theory has proven itself to be useful in the study of the totality of mathematical systems of a given type. In this paper we shall continue one of such studies by investigating further the lattice of all topologies on a given set 5. A considerable amount of research has been done in this field (1; 2; 3; 5; 6). This research, besides satisfying the intrinsic interest in the lattice theoretic properties of this lattice, has aided the study of interconnections of different
more » ... ons of different properties of point set topologies. We shall show that the lattice of all topologies on a set consisting of more than two elements has only trivial homomorphisms. On the other hand it will be shown that this is not true for the lattice consisting of all JYtopologies on S and the lattice of complete homomorphisms will be constructed in this case. We shall also show that the lattice of all topologies is complemented if 5 is finite. Finally we shall construct the group of automorphisms for the lattice of all topologies and for the lattice of all TVtopologies on S. We shall conclude with a definition of a lattice theoretic property which clarifies the change of properties of the lattice of topologies as we go from the finite to the infinite case. We shall represent a topology R on the set 5 by the collection of its closed sets, R -\S a }. Il Ri and R 2 are topologies on S then Ri < R 2 if and only if every set closed under Ri is also closed under R 2 . It can be seen that under this ordering the set of all topologies on S forms a complete point lattice. The intersection of two topologies Ri and R 2 in the lattice is the topology whose closed sets are the sets closed under Ri and R 2 . The union of two topologies Ri and R 2 is the topology whose closed sets are intersections of finite unions of the closed sets of Ri and R 2 . Let us denote the lattice of all topologies on 5 by LT(S) and similarly let LTi(S) denote the lattice of all 7Vtopologies on S. We shall now investigate the homomorphisms of LT(S). LEMMA 1. If 6 is a nontrivial homomorphism on a point lattice L, then there exists a point p of L such that p = 0(6). Proof. Let 0 be a non trivial homomorphism on L. Then there exist two elements a and b in L, a > b, such that a = b(6). Since L is a point lattice there exists a point p such that a Pi p = p and b P\ p = 0. But then p = a np = br\p = o.
doi:10.4153/cjm-1958-054-5 fatcat:z636lalotrdf7ag2srjskivc5a