The two notions of continuity (sequential continuity and topological continuity) you present under “Multivariate continuity” are not equivalent. In a sense the topology around a point can be ‘too large’ to recover it from just convergence of sequences (in particular, these notions are equivalent for first countable spaces (I think? Second countability is definitely enough, but I think first countability also is) but not for general topological spaces). You can fix this by replacing the sequences with nets.
The compactifications (one-point and Stone-Cech) are very useful for classification and representation theorems, but personally I’ve hardly ever used them outside of that context. These compactifications are very deep mathematical results but also a bit niche.
I remember back when I took my course on Introduction to Topology that we spent a lot of time introducing homotopies and equivalence classes, and later the fundamental group. And then all that hard work paid off in a matter of minutes when Brouwer’s fixed point theorem (on the 2-dimensional disc) was proven with these fundamental groups, which is actually one of the shorter proofs of this theorem if you already have the topological tools available.
Very nice! Two small notes:
The two notions of continuity (sequential continuity and topological continuity) you present under “Multivariate continuity” are not equivalent. In a sense the topology around a point can be ‘too large’ to recover it from just convergence of sequences (in particular, these notions are equivalent for first countable spaces (I think? Second countability is definitely enough, but I think first countability also is) but not for general topological spaces). You can fix this by replacing the sequences with nets.
The compactifications (one-point and Stone-Cech) are very useful for classification and representation theorems, but personally I’ve hardly ever used them outside of that context. These compactifications are very deep mathematical results but also a bit niche.
I remember back when I took my course on Introduction to Topology that we spent a lot of time introducing homotopies and equivalence classes, and later the fundamental group. And then all that hard work paid off in a matter of minutes when Brouwer’s fixed point theorem (on the 2-dimensional disc) was proven with these fundamental groups, which is actually one of the shorter proofs of this theorem if you already have the topological tools available.
Wrt continuity, I was implicitly just thinking of metric spaces (which are all first-countable, obviously). I’ll edit the post to clarify.