While the topic of topos subobject classifier is current, how far back can the historical and conceptual origins of the use of "$\Omega$" as the symbol denoting such be traced?
Regards, Keith Harbaugh
If a topos E is a generalized topological space, then Omega is the internal version of its (ungeneralized) topology, its lattice of opens: for the global elements 1 -> Omega are in bijection with the continuous maps [geometric morphisms] from E to [sheaves over] Sierpinski space. (This extends to generalized elements too. Suppose U is an object of E, a sheaf over E, and let U' -> E be the corresponding local homeomorphism. Then the morphisms U -> Omega correspond with the opens of U'.) Hence Omega can be read as an abbreviation for Open. Is this how the notation actually arose, or is it just a pleasant rationalization of my own? If X is an ordinary topological space, then Omega X is one notation used for its topology. But though I use it myself, I don't know anything about its history. Putting these together you see that when you take E as your base category of sets then Omega 1, the topology of the 1-point space, is just Omega. Steve Vickers Omega University