categories: Re: the alpha of Omega

[S Vickers <s.j.vickers@open.ac.uk>]

>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