Post Made Community Wiki by Harry Gindi
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Well, this isn't a full answer, but I think it's worth posting.

We can identify a Hausdorff space with its poset of open sets because every convergent (thanks JDH) ultrafilter converges uniquely to a point, so in fact, all of the "set data" of the space is contained in the poset of opens itself. We can define a lattice structure on this poset that takes the place of our algebra defined by intersections and unions. This is why the category of Hausdorff spaces is actually a category monadic over sets. Hausdorff spaces are actually totally described by their algebras, which seems pretty cool to me.

Some speculation on the question:

We have a natural poset structure on subobjects of algebraic objects, which at least gets us partway to having a topology.

Further, we can classify maps out of the space by looking at kernels of maps. There is also a canonical action for normal subgroups and ideals (product of normal subgroups and sum of ideals). These have the nice property that they are closed under this operation. They are also closed under intersections. This gives us a complete modular lattice on kernels. The interesting case about rings is that we have a third operation, the product of ideals. The interesting thing about the product of ideals is that it is only defined for finite products. Then we have a somewhat natural structure to start working in (at least for rings).

I believe what Qiaochu was talking about with the Galois connections is that for any algebraic structure, you can associate this poset structure on subobjects, and even better, sharpen the characerization by looking at kernels (at least in the case of groups and rings). However, my point was that the additional operation of multiplication, which is restricted to finite products (I guess I would say it has finite arity, but that's not exactly right), gives us an operation on the poset that looks like "finite unions" or "finite intersections".

3 added 554 characters in body

Well, this isn't a full answer, but I think it's worth posting.

We can identify a Hausdorff space with its poset of open sets because every ultrafilter converges uniquely to a point, so in fact, all of the "set data" of the space is contained in the poset of opens itself. We can define a lattice structure on this poset that takes the place of our algebra defined by intersections and unions. This is why the category of Hausdorff spaces is actually a category monadic over sets. Hausdorff spaces are actually totally described by their algebras, which seems pretty cool to me.

Some speculation on the question:

We have a natural poset structure on subobjects of algebraic objects, which at least gets us partway to having a topology.

Further, we can classify maps out of the space by looking at kernels of maps. There is also a canonical action for normal subgroups and ideals (product of normal subgroups and sum of ideals). These have the nice property that they are closed under this operation. They are also closed under intersections. This gives us a complete modular lattice on kernels. The interesting case about rings is that we have a third operation, the product of ideals. The interesting thing about the product of ideals is that it is only defined for finite products. Then we have a somewhat natural structure to start working in (at least for rings).

I believe what Qiaochu was talking about with the Galois connections is that for any algebraic structure, you can associate this poset structure on subobjects, and even better, sharpen the characerization by looking at kernels (at least in the case of groups and rings). However, my point was that the additional operation of multiplication, which is restricted to finite products (I guess I would say it has finite arity, but that's not exactly right), gives us an operation on the poset that looks like "finite unions" or "finite intersections".

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