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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.

Tom Kacvinsky
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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.

Nick
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  • Ah, thank you. It turns out the professor I was talking to sent me an excerpt from J.L. Kelley's book about the BRT but there it talked about using Lindelöf spaces, not second countable spaces. But looking up second countable, that's almost a Lindelöf space, as Lindelöf spaces are such that each open cover of the space has a countable sub-cover, wheres second countable spaces just have a countable base. Need to look up the theorems between Lindelöf and second countable spaces. It's been over twenty years since I received my masters and I am looking to get back into continuum theory. – Tom Kacvinsky May 23 '21 at 22:04