How to develop intuition in topology? Is there any efficient trick (besides doing exercises) to develop intuition in topology?

The question is general but i would like to add my view of things.

I started to teach myself topology through several books a couple of months ago. I already passed the point of being overwhelmed by the amount definitions. Most of them I remember although sometime i check to remind myself (I have, after all, a terrible memory). 

The point is most of the theorems and exercises I prove don't sink in and usually whenever I'm given a statement to prove I start with the definitions and work up from there. My feeling is that it's part of the nature of the subject. Browsing through Counterexamples in Topology really makes my head turn (all the one way implications... what ever happened to "if and only if"?)

This is in contrast to when I’m doing problems in analysis where i have a visual picture which tells me usually straight away if a given statement is true or false even before i start proving it.

I think that time here is a key element and intuition will inevitably develop at some point and so my question is: 

Is there any efficient way to develop intuition in topology?  

By intuition I mean a mental model that helps you see things more clearly for example: 
If you’re given a space with certain properties (say first countable, countably compact hausdorff space) than your intuition tells you it has to have some other properties ($T_3$ in this case).
In my opinion, one of the best ways of developing intuition in topology is to study other branches of mathematics in which there are topological spaces. Many of these definitions, properties and theorems were imagined by people who were working in related branches of math, mostly analysis and geometry. These people stumbled upon spaces which had remarkable or singular properties, so they studied these properties. The examples came first.

Why does anyone care about compactness, say? The best way to answer this question is to ask the question: who were the first people to care about compactness, and why did they? What kind of spaces were they working with?

If you want to learn a new language, there is no point in reading the dictionary and the thesaurus. There is also not much point in learning all of the bizarre exceptions before you encounter them naturally. Instead, you should learn a few basic principles, and then go out and talk to people. Figure out how they speak, and refer to the thesaurus as you go along.
The way I developed intuition in topology is from advanced multivariable calculus. All the concepts of "open", "closed", "compact", "convergence", "connected", and so forth, have a very visual meaning behind them when you study these concepts from advanced calculus. So when you see the abstract definitions in topology they are similar to how it comes up in advanced calculus.