I am teaching undergraduate topology this semester; you can see some stuff that I’ve posted for undergraduates here.
I took the “start with metric spaces and topology of approach and am going slower than I’d like. But it takes some time to absorb the stuff.
So, we are finally gotten up to homeomorphisms (still in basic metric spaces) and so I figured that we were finally ready to show something like: is not homeomorphic to (the unit circle). Fine: suppose a homeomorphism exists and decompose where
Not a problem so far…so now pull back the disjoint open sets to the unit circle minus one point…and….then…I ….realized….that…I have not proven that the interval is a connected set; in fact I haven’t even defined “connected set”. &^%$#. Now, that isn’t that hard to do, but it does take time and one has to do some setting up.
And that is one problem with teaching this course: you have to do so much elementary stuff to even begin to prove something elementary.