So, this is definitely a soft question and I apologize. I've been in point set topology for about a week and I have two questions,
Everything spews from definition, so should I dismiss my geometric intuition?
Are definitions meant to be taken as fact, because mostly they seem to come from nowhere?
I am using Munkres. So far I like topology, but the abstraction and my usual visualization isn't working. Should I just pound the definitions into my brain?
Please just comment, i realize this may be really hard to answer, so once a few comments come in (if any), I'll delete the post.
Definitions do not come from nowhere, but it is unfortunately conventional to omit the rationales behind them.
In topology, popular accounts say that the subject studies properties of spaces that don't change if one stretches and bends them without tearing the. But then rigorous accounts say that a topology is a collection of subsets, called open sets, satisfying certain conditions. It seems to me that anyone who doesn't wonder why those are the same thing is not paying attention.
Here's a simple example. Consider the half-open interval $[0,1)$. Let the "basic" open sets in this space be the intersections of $[0,1)$ with open intervals, and let open sets be arbitrary unions of "basic" open sets. Then one has an interval that looks like an interval. But then alter the definition of "basic" open sets so that every open set containing $0$ as a member includes some set of the form $[0,\varepsilon)\cup(1-\varepsilon,1)$, so that both ends of the interval are included. Then one has an interval that looks like a circle. In other words, the way the whole thing is connected together is just a matter of which sets are open. That enables you to see why the intended meaning of topology is the same as the meaning according to the logically rigorous definition.