Love this question and all the answers. I find topology fascinating even though I understand maybe 5% of it. I took a decent amount of math as part of my CS degree, but beyond basic calculus it was concentrated in probability, stats, and linear algebra; never came near topology. In hindsight I wish I had taken more math, but as a 19-20 year old student at the time, I was happy to be done with it.
If I invented a time machine, sometimes I think my second use of it would be to give my college self class choice and scheduling advice.
Topology is fun, and not especially mysterious if you give it the time. One of those subjects that is at the same time highly abstract and exercises the visual–spatial–kinetic thinking part of your brain. You can learn it whenever you like! (For a motivated student, I think self study of mathematics topics is better than a course with fixed problems anyhow, because you can move at your own pace, and take any path you prefer. Requires some focus though.) I don’t have enough experience with all the various textbooks to compare them, but I thought Munkres was alright.
It depends, modern algebraic topology is frequently considered to be one of the most mysterious and abstract fields of math, along with things like algebraic geometry.
I wouldn't compare algebraic topology to algebraic geometry. In my opinion the later is much more difficult both one the conceptual and technical side.
Really? I don't find conceptual and technical difficulty of considering (co)homology theories determined by spectras/infinite loop spaces like K-theories and cobordism theories, study of stable homotopy theories, spectral sequences, triangulated categories etc. to be much smaller that the difficulty of algebraic geometry. As I said, I find them both to be really abstract and mysterious.
Many concepts from algebraic topology can be applied to algebraic geometry after appropriate redefinitions. These new tools seem harder to use than originals. For example I find it technically more difficult to work with étale fundamental group than with a topological fundamental group. Similarly cohomology of a topological space looks simpler than cohomology of a scheme with coefficients in a sheaf. There are many similarities but working with schemes and sheaves is more troubling (at least for me). This of course can follow not from intrinsic difficulty of the subject but from my ignorance and inappropriate intuitions.
I majored in math with a minor in CS (undergrad). Of his entire technical explanation, I'm only familiar with the notions "automorphism" and "topology". To add insult to injury, I did have a course on topology, but it covered only so-called "point-set" topology, and not the algebraic variety thereof.
If I invented a time machine, sometimes I think my second use of it would be to give my college self class choice and scheduling advice.