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).
 A: 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.
A: Read "Explorations in Topology" by David Gay. I'm too in search for understanding in topology and this was recommended by my adviser (topologist). So far the book is more than expected. It's amazing. 
A: Though I don't have any authority on this specific subject matter, I would say the thought of looking for a "trick" or "shortcut" for anything is quite dangerous. Just put the head down and work laboriously is the best approach to everything. "A couple of months" is obviously extremely short time. To really gain any basic understanding about anything you would take at least 3 years or even more. To master one? Probably 10 years or more. In everything I learned, "exercises" are probably exactly the thing that helped me gain "intuition", if any. If you always think for "shortcut" or "trick" you can very well be unintentionally slowing yourself down by distracting yourself too much. This is just a reminder; hopefully I won't be bashed too much for this unanswerlike answer :P
I also don't agree with Joyal's perspective on learning languages. Of course you go and talk to people. But in fact mastering the usage of words and special instances through enough exercises alone will immensely speed up the process. That's "achieve multiple times of effect in half the time" in Chinese. If you don't have a decent command of quite a few words and phrases you're never gonna understand anything no matter how much you "talk". That's why paying unbalanced attention at any of those both aspects will lead to disasters.
A: I found this idea of a "Lattice of Topologies" and it really helps me to visualize the relationships between different properties. I now think of different topologies on set as living on this lattice somewhere.
This MO question is where I found the term. The question is interesting and helpful in it's own right.
A: The best apparatus for use in investigating topology are the definitions of various terms in the statement of a theorem, previous results, and the power to discover their natural logical connections. To gain intuition is simply a dream that will never come true.   
A: Let's do an example: let's say we want to know when limits are unique in a topological space. Here's the proof of the theorem in a metric space:

Let $(a_n)$ be a sequence with limits $L_1$ and $L_2$. Then $a_n$ is eventually within every neighbourhood of $L_1$ and every neighbourhood of $L_2$. If $L_1\ne L_2$, we can choose the neighbourhoods to be disjoint. Contradiction.

This is completely equivalent to the proof you've probably seen, but I've phrased everything in terms of neighbourhoods, which are fundamentally topological concepts. The only fact we used is the existence of disjoint neighbourhoods of distinct points. Limits being unique is pretty important, so we call a space where distinct points allow disjoint neighbourhoods a Hausdorff space or T2 space.
(It's also worth thinking about why the generalisation goes for  limits of nets, rather than limits of sequences)
The trick I'm suggesting is to "work backwards" from theorems you can tell are important (as opposed to some inane statement about open sets): (1) start with an important theorem in analysis, (2) go through its proof, (3) work out what axioms you need and simplify them to a form involving just open sets.
Some more examples of such generalisations:


*

*Every open neighbourhood of a limit point of $S$ contains an infinite number of points in $S$. (T1 space)

*Finite sets are closed. (T1 space)

*Continuous extension theorem. (T4 space)

*Bolzano-Weierstrass theorem (compact sets)

*Intermediate value theorem (connected sets)


You may find this series of articles I wrote illuminating to this end: https://thewindingnumber.blogspot.com/p/2204.html
A: 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. 
