In my last post, discussing topologies and negation, we were concerned with the use of the open sets of a topology as a partial order over which it was possible to define an algebra (a complete Heyting algebra, in fact). In this context, the underlying set on which the topology is defined is of less interest than the lattice of its open sets - once we have the shape of that lattice, the “points” of the topology (the elements of the underlying set) more or less recede into insignificance. We don’t really care what’s “in” each open set: what we care about is how the open sets stand in relation to one another. I’m now going to discuss some of the ways in which the “points” of a topology can be exposed or obscured within the lattice of its open sets.
We can define an equivalence relation on a topological space (X, T) as follows: for all x and y in X, x and y are equivalent if every open set in T that includes x also includes y. In other words, x and y are equivalent if they are indiscernible on the basis of inclusion in the open sets of T. We now define a T-0 topology as one in which, for any two elements x and y of the underlying set (where x != y), there is at least one open set that contains either x, or y, but not both. In other words, the open sets are such that no two different “points” of the topology are equivalent according to the equivalence relation just given.
Take the topology (X, T) where T = {{}, X} - that is, where the open sets are the empty set and X itself. In this topology, all of the elements of X are equivalent with respect to T. For any pair of elements x, y of X, the empty set by definition includes neither and X by definition includes both. This is not a T-0 topology. At the other extreme, the discrete topology (X, T) where T = P(X) has the “singleton” open sets {x} and {y} for any x and y in X. No two (non-equal) x and y are ever equivalent with respect to this topology. We can say that all of the “points” of T are completely exposed, or that the elements of X are fully discernible in T.
For any set X = {0, 1, 2, 3…}, the most coarse T-0 topology (the one with the fewest open sets) is one that “counts” the elements of X in its open sets like so: {}, {0}, {0, 1}, {0, 1, 2} all the way up to X. We can see that for any pair of elements x and y, where x < y, there is an open set that “counts up to” x but not to y. All of the T-0 topologies on X lie somewhere between this topology and the discrete topology on X; that is, they can be reached by adding open sets to the coarsest topology, or by taking open sets away from the finest topology in such a way that all of the open sets of the coarsest topology still remain.
(Remember that when adding or removing open sets, we must ensure that the topology remains closed over union and finite intersection - for example, if we add the singleton {2} to the open sets {}, {0}, {0, 1}, {0, 1, 2} etc., we must also add the set {0, 2}, since that is the union of {0} and {2})
One interesting feature of the most coarse T0 topology is that the negation of every open set in it other than {} is {} - and the negation of {} is the maximal element X. This has the consequence that while not-not-P is only P for the minimal and maximal elements, not-not-not-P is always equal to not-P and not-not-not-not-P is always equal to P.
(Readers of Logics of Worlds may want to know whether all of this has anything to do with the theory of “points” laid out in the later sections of that book. My answer at this point is that I don’t know yet).