Brouwer Reduction Theorem

Question

I went looking for a statement of the Brouwer Reduction Theorem, but Google only gives hits for his fixed point theorem. I talked to an old professor of mine about it being used to prove if you have two mutually exclusive closed sets, there exists and irreducible continuum that intersects each. Once I have a statement of the BRT, I think I can prove the the above, and then work on proving the BRT.

Answer 1

Brouwer’s reduction theorem: If $F$ is a closed subset of of a second countable topological space $X$ and $F$ possesses an inductive property $P$, there is an irreducible closed subset of $F$ which possesses $P$. A property $P$ of subsets of $X$ is called inductive iff whenever each member of a countable nest of closed sets has $P$, then the intersection has $P$. Also a set $F$ is irreducible with respect to $P$ iff no proper closed subset of $F$ has $P$.

Edit: A proof is here.