It's very hard to appreciate compactness as a concept in its own right in a context where it just means "closed and bounded."

If this was really the rule, I never would have learned any analysis at all. In the context of analysis there were simply too many details floating around for me to see what was relevant and what wasn't. It made next to no sense to me until I learned a bit of topology, which set aside the parts that don't matter and let me focus on what was actually relevant. At that point I was able to go back and learn the analytic concepts quite easily, but without the chance to study general topology I don't think I could have.

Re: the quote from Rudin, I once graded a real analysis course using Wade's text, which essentially proceeds by saying "assume there exists a complete ordered field" and then proving theorems that hold in such an object. I had to tweak the way I graded the course since students didn't really appreciate that perspective on analysis at that stage in their development.

A rare example of a family where the granddaddy is the youngest...

"In fact, one can define the homology groups of a space as the homotopy groups of its infinite symmetric product (= free topological abelian monoid on the (pointed) space)." Is there a good reference for this approach?

So at the time the University of Chicago had a test you could take that guaranteed you admission to grad school?

I take it "gatti" is the Italian word for "cats"?

Well, if X is a set, then the Cartesian product $X^n$ is the set of functions from $[n] = {1, 2, \ldots, n}$ to X, so naturally $X^A$ is the set of functions $A \to X$.

For fellow monolinguals: "complessive" is apparently an Italian adjective which has not yet been imported into English, and which means something like "comprehensive."

This book is probably a large part of why I wanted to be a mathematician when I grew up.

I think he's talking about "spitten" (past participle) versus "spitting" (present participle).

When I was shown the Reidemeister moves in school, several of my classmates and I made the objection, in essence, that it wasn't clear that they generated everything. Worse, since we didn't have topology to work with, we didn't really have a "real" definition to compare it with, so it felt to us that the real issues were being swept under the rug.