What is the big picture of topology I'm a big picture person, and usually feel like I'm swimming in formulas if I don't see the big picture of a branch of math.
If someone were to say to me "what is the big picture of differential calculus?" I would tell them that when provided with a function, find its rate of change everywhere (that it is continuous and differentiable in the first place) so that when given a particular x value, you can immediately find the function's rate of change at that point, should you need it to solve some bigger problem.
If someone were to say to me "what is the big picture of differential equations?" I would say that when provided a differential equation, find some way to extract an algebraic equation that provides a solution to the equation everywhere (or everywhere that the equation is defined) so that you don't have to keep solving it over and over again.  Or, at the very least, show that the solution to the ODE can at least be converted into some other, already solved differential equation.
I realize that this is oversimplified, but I'd call these the big pictures of these fields.
What then, is the big picture of general topology?
I'm sure this question will generate a lot of claims that the question is invalid.
 A: Actually there is probably no big picture of any mathematical subject, but rather a huge collection of tiny puzzle pieces. If you are blessed, you may reach a level at which you can see a small fraction of the "big picture".
Just to give an example regarding calculus. You can learn calculus all your life, and even teach it and write books about it all your life, (the 19th edition of which coming out shortly so that no college will use the previous one), and you think - you hope -- you did get it, you did grasp the big picture of calculus, and then you come across a little book from more than 50 years ago by Michael Spivak, called "Calculus on Manifolds", and you suddenly realize you were only gazing at a small part of the big picture. Just one example of many.
A: For whatever it's worth I think this is a good question, if by "big picture" you mean that you want an intuitive feel for questions like "What are we doing here [in point-set topology]?" and "Why are we making these definitions?". I will strongly disagree with the notion that there is no "big picture" of any mathematical subject, as I understand the term "big picture".
If you find it natural to think in visual, spatial, geometric terms, then point-set topology (or general topology) has a very natural interpretation: point-set topology is a game where you try to write down definitions of concepts that are intuitively obvious in $\mathbb{R}, \mathbb{R}^2$ and $\mathbb{R}^3$ without ever writing down the word "distance".
Examples of topological generalizations of intuitive concepts:

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*An open set $U$ is, informally, a set such that no matter where you are inside $U$, you can move some distance in any direction and still remain inside $U$. This is obviously compatible with the intuitive understanding of an open interval in $\mathbb{R}$ or an open disc or ball in the higher-dimensional spaces. In point-set topology we evade this mention of distance with the use of three axioms which you can look up, which again make sense in the basic spaces.

*In a metric space, the limit of a sequence is a point such that for any distance from the limit, only finitely many points in the sequence are not within that distance from the limit. In topology we re-define this by requiring that for any open set $U$ containing the limit, only finitely many elements of the sequence are not in $U$. Intuitively you can imagine considering smaller and smaller open sets $U$ and you see why this distance-free definition of a limit is sensible.

*The separation axioms provide a variety of answers to the question "To what extent are two points allowed to be in the same place?" In the basic spaces I mentioned this is an absurd question, but topology allows us to consider more abstract spaces that can still be useful. Perhaps the most important separation axioms is the Hausdorff or $T_2$ axiom, which requires that distinct points $x$ and $y$ in the space can be enveloped in their own respective open sets $U$ and $V$ such that $U \cap V = \emptyset$. In words, any two distinct points can be put inside their own open regions which do not overlap at all. The basic spaces I mentioned are obviously Hausdorff.

Why do we do this? What are the benefits of this abstraction?

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*The theory developed clarifies and simplifies results from related disciplines. For example, there is a result in real analysis stating that a continuous function on closed interval has a maximum value. Non-topological proofs of this result can be quite messy in my experience, but topology offers a cleaner proof that I find is easier to understand and extract meaning from.

*The theory developed, due to its abstract nature, is applicable in spaces other than the basic spaces I mentioned, and it turns out that study of these abstract spaces can be very profitable. The Zariski topology, having to do with zeros of polynomials, is fundamental in algebraic geometry, one of the major areas of modern mathematics. The theory of manifolds is inherently a topological theory, since manifolds are defined to be particular topological spaces, and manifolds are pervasive across geometry and modern physics.

*A bit more vaguely, it's my opinion that topology provides a great training ground in the sense that study of it imparts two lessons: the power of abstraction (especially when paired with concrete, intuitive examples), and the importance of finding good definitions and the "right" objects to study. I am not an advanced mathematician by any stretch, but I feel that these lessons are important for anyone trying to get a feel for advanced math.

Lastly, the picture on this Wikipedia page is very helpful in my opinion because it immediately gives you a basic understanding of the relationships between four types of spaces that are common in analysis, at least. Topology underlies all of analysis, and probably all of geometry too.
A: I'm definitely not an expert in topology, but, I have read that  topology is considered to be the study of continuity, broadly construed.
