TITLE: How to think/see point-set topology abstractly?
QUESTION [5 upvotes]: I've started learning point-set topology this semester.  I've learned basic material about:

topology on a set
topological space
open sets
closed sets
clopen sets
closure
neighborhoods
interior point
interior
exterior
boundary
boundary point

However, whenever I think of terms such as neighborhood, interior point, boundary $\dots$, I tend to think of them in terms of $\mathbb R^2$. (e.g., circles on the plane)
Is there a way for me to think of these, and future topology terms, abstractly?Intuitively?
What do you see in your mind, or think about, when you hear these terms? How do you look at this?

I like to give this example from abstract algebra for what I mean when I say intuitively:
If we have a group and we mod out by the commutator subgroup, what's basically going on is that we are setting all the elements that do not commute equal to the identity and thus we are left with a quotient group where everything commutes.

REPLY [2 votes]: General Topology is loosely a generalization of geometry, so I would say that the best way to get a grip on (General) Topology, and learn to get to get intuition for General/Point-Set Topology, is to learn how to visualize what is going on. 
And the best way to visualize what's really going on, is to draw pictures of the definitions and concepts that frequently pop up.
For the last few concepts of neighbourhoods, interior point,
interior,
boundary,
boundary point, these are all really visually intuitive if you draw pictures of them out. As an exercise try to come up with drawings to encapsulate these definitions.
For example a normal space is a topological space, where for every two disjoint closed sets, $E$ and $F$, there exists disjoint open $U$ and $V$ containing $E$ and $F$ respectively.
This can be easily encapsulated by the following drawing (courtesy of Wikipedia)

Finally I should add, that drawing pictures is merely a means to gain intuition, but it is something that you should not shy away from.