I'm a first year grad student. After listening to my algebra professor's explanation that for any finite set A, P(A) is a vector space under the operation of symmetric difference over the field . It got me to wondering what a metrization of the power set might look like. So, I came up with:
Where d is defined to be zero if both U and V are the empty set, and I haven't formulated anything for infinite sets, yet.
This seems like it would, indeed, be a metric on P(A), but it induces the discrete topology, obviously. (Well, it wasn't obvious to me, but it was to all of my professors who found my efforts entertaining until I discovered that I was attempting to prove the existence of a metric topologically equivalent to the 0-1 metric). I haven't actually proven the triangle inequality yet, but it seems likely that I could if I expended the effort.
Anyway, it wasn't a total waste of time. I learned quite a bit from the exercise. And I would like to continue my efforts. So, I'm reevaluating my strategy. I thought I would start with something a little simpler, which I know more about, but for which my professors don't believe a metric can be found which has a topology more interesting than the discrete one.
Rather than an arbitrary power set, I thought I would try:
Now, I have a lot more things I can try to play around with. For instance, some form of sparseness function that calculates maximal element - minimal elemnt and divides by cardinality? I haven't decided yet how it might work, but that seems to work under unions (where the sparseness of the union would be less than or equal to the sum of the sparseness for the separate sets). I made up the word sparseness, of course, but I haven't thought about it enough to give it a decent term.
Anyway, any ideas for how I might proceed would be greatly appreciated! Does anyone know if I can norm a power set in such a way that it can be metrized with the induced topology anything other than discrete?