I was reading the blog topological musings, and the category talk reminded me of this neat thing I posted on AoPS awhile ago.
By using the Mayer-Vietoris sequence in cohomology, we determine whether or not can be written as the union of two open connected sets, U and V, such that is disconnected.
Well, it seems that we are concerned with stuff, since that tells us the number of connected components.
The sequence gives , a nice short exact sequence in which we mostly know they are connected, so
, basically the question reduces to: Is it possible for the second to last term to have dimension greater than 1, but the sequence to remain exact?
Well, no, since the exactness tells us the dimension of is the dimension of .
So we have a nifty result: the plane cannot be broken into two connected parts in which the intersection of these two parts is disconnected.
Challenge for the readers: This seems extremely simple and obvious. Find a proof that does not require advanced techniques such as cohomology.