Being able to write and explain the complete proof of the Banach-Tarski theorem is a good proxy for understanding the basics of topology: it appears in the middle of most intro topology books for this reason.
What? A large part of the proof, as I understand it, is basically group-theoretic. Most of the central concepts of topology don’t seem to appear. Are you thinking of a different theorem?
Hmm, wikipedia gives a very different proof than the one I learned for Banach-Tarkski, but it’s also possible I’ve misremembered and I was thinking of Tychonoff’s theorem.
I’d be astonished if there were a proof of Banach-Tarski that uses (1) most of the key notions in basic topology and (2) not much else. And I’ve certainly never seen a proof of BT in the middle of an introductory (or any other) topology book.
Being able to write and explain the complete proof of the Banach-Tarski theorem is a good proxy for understanding the basics of topology: it appears in the middle of most intro topology books for this reason.
What? A large part of the proof, as I understand it, is basically group-theoretic. Most of the central concepts of topology don’t seem to appear. Are you thinking of a different theorem?
Hmm, wikipedia gives a very different proof than the one I learned for Banach-Tarkski, but it’s also possible I’ve misremembered and I was thinking of Tychonoff’s theorem.
I’d be astonished if there were a proof of Banach-Tarski that uses (1) most of the key notions in basic topology and (2) not much else. And I’ve certainly never seen a proof of BT in the middle of an introductory (or any other) topology book.