What is a topological space good for? I know there are already some questions similar to this, which all give an answer that a topological space creates some structure on a set which is an abstraction of distance and makes it possible to define other concepts like connectedness, compactness, metrizability asf. I understand this in theory, but when I create a simple example I fail to grasp what distance/closeness means in a topological space.
For example, when I have the following set $X$ and a set of subsets $\tau_1$ which does not satisfy the axioms of a topological space:
$$ X=\{a,b,c\},\tau_1=\{X,\emptyset,\{a,b\},\{b,c\} \} $$
now I make this into a topological space by adding the sets $\{c\}$ and $\{b\}$
$$ X=\{a,b,c\}, \tau_2=\{X,\emptyset,\{a,b\},\{c\},\{b,c\},\{b\} \} $$
What is "better" now and how is this related to some notion of distance or closeness?
In other words, what did I gain by adding the subsets $\{c\}$ and $\{b\}$ to $\tau_2$? 
More over what can I do with $\tau_2$ I can't do with $\tau_1$?
 A: I like to think of topological spaces as defining "semidecidable properties". Let me explain.
Imagine I have an object that I think weighs about one kilogram. Suppose that, as a matter of fact, the object weighs less than one kilogram. Then I can, using a sufficiently accurate scale, determine that the object weighs less than one kilogram. Even if the object weighs, say, 0.9999996 kilograms, all I need to do is find a scale that's accurate to within, say, 0.0000002 kilograms, and that scale will be able to tell me that the object weighs less than one kilogram.
This means that "weighing less than one kilogram" is a semidecidable property: if an object has the property, then I can determine that it has the property.
Suppose, on the other hand, that the object actually weighs exactly one kilogram. There's no way I can measure the object and determine that it weighs exactly one kilogram, because no matter how precisely I measure it, it's still possible that there's some amount of error which I haven't discovered yet. So "weighing exactly one kilogram" is not a semidecidable property.
What does this have to do with topological spaces? Well, an open set in a topological space corresponds to a semidecidable property of that space. This is why in the topological space of real numbers, the set $\{x : x \in \mathbb{R}, x < 1\}$ is an open set, but the set $\{x : x \in \mathbb{R}, x = 1\}$ is not.
So, consider the "topological space" $X = \{a, b, c\}$ with open sets $\emptyset$, $\{a, b\}$, $\{b, c\}$, and $\{a, b, c\}$. In this "topological space", you are asserting that


*

*(since $\{a, b\}$ is open) if you have a point which is either $a$ or $b$, then it is possible to measure it and determine that it is either $a$ or $b$ (though it is not necessarily possible to determine which one it is);

*(since $\{b, c\}$ is open) if you have a point which is either $b$ or $c$, then it is possible to determine that it is either $b$ or $c$; but

*(since $\{b\}$ is not open) if you have the point $b$, it is not possible to determine that it is $b$.


However, these assertions contradict each other. Suppose that you have the point $b$. Because of the first bullet point, there is some measurement you can make which will tell you that the point is either $a$ or $b$. And because of the second bullet point, there is another measurement you can make which will tell you that the point is either $b$ or $c$. If you simply make both of these two measurements, then you will have successfully determined that the point is (either $a$ or $b$, and either $b$ or $c$)—in other words, that the point is $b$. But the third bullet point asserts that this is impossible!
For more explanation of this idea, see these two answers:


*

*https://math.stackexchange.com/a/31946/13524 (Qiaochu Yuan's answer to "What concept does an open set axiomatise?")

*https://mathoverflow.net/a/19531/5736 (community wiki answer to "Why is a topology made up of 'open' sets?")

A: As usual in mathematics, the advancements and enrichments of the theory make so that the definitions become increasingly more simple, but their usefulness not so clear.
Firstly, the category of topological spaces behave extraordinarily well with respect to inducing structures. For instance, it is not clear which metric to take on a product of metric spaces (even finite), but it is clear what topology to take.
More explicitly, we have the following neat characterizations of topologies:


*

*Given topological spaces $X,Y$, the product topology on $X \times Y$ is the smallest topology which makes the projections continuous. (this also holds for infinite products)

*Given a topological space $X$ and a subset $A$, the induced topology on $A$ is the smallest topology which makes the inclusion continuous.

*Given a topological space $X$ and an equivalence relation $\sim$ on $X$, the quotient topology on $X/\sim$ is the biggest topology which makes the projection continuous.

*The topology induced by a metric on $X$ is the smallest topology which makes $d$ a continuous functions.

*etc


And such characterizations are useful in other areas of mathematics. For instance,


*

*The weak topology is the smallest topology on a Banach space for which the dual still consists of continuous maps.

*The weak-star topology is the smallest topology on a dual of a Banach space for which the elements of the standard embedding of it on its bidual are still continuous maps.


Note, for instance, that there is no natural way to induce a metric on a quotient. In fact, in a lot of examples above, a metric can't be induced that would give the topology requested.
You should not try to understand topology by means of "distance". This is akin to understanding magnets in terms of rubber bands. It is an entire new concept, and you must come in terms to it as it is. Sure, analogies are fine sometimes, but that is as far as they go. They won't provide deep understanding (maybe comfort, though), and trying to grasp to them at all times can be a huge hindrance.
I'm afraid there is not much I can do other than exemplify as I did above to prove that topologies are useful. Nevertheless, it may be useful to say that topologies are to continuous maps as groups are to homomorphisms (quite literally). Metric spaces are not the best ground for continuity, they are not the natural ambient setting. 
As a sidenote, a lot of useful spaces are not metrizable. For example, the weak-topology and weak-star topology are not metrizable in general, neither is the test function space on an open subset of the Euclidean space. 
But even if you are dealing with metrizable spaces, treating them in terms of topological spaces can be very fruitful (due to the fact that inducing metrics is not a trivial matter). For instance, we often want to treat the torus as a square with sides glued in an appropriate manner. This has a very simple topological description, but not an obvious metric one.
To quote Bredon in the beginning of his "Topological Spaces" section:

"Although most of the spaces that will interest us in this book are metric spaces, or can be given the structure of metric spaces, we will usually only care about continuity of mappings and not the metrics themselves. Since continuity can be expressed in terms of open sets alone, and since some constructions of spaces of interest to us do not easily yield to construction of metrics on them, it is very useful to discard the idea of metrics and to abstract the basic properties of open sets needed to talk about continuity. This leads us to the notion of a general 'topological space'."

A: There is no natural distance in an abstract topological space, when there is one it is call a metric space.
The thing about a topological space is that you can take finite intersection and still remain in the topological space. For instance, with $\tau_1$, when you are interested about $\{b,c\}\cap\{a,b\}$, you don't get an open set, which isn't practical. This doesn't happen in a topological space like $\tau_2$.
Basically the purpose of topological space is to have certain properties and theorem, based on the axioms of the space. If you change the axioms, the usuals properties you always use won't work anymore. So in order to preserve what has already been proved, you must work with the rights axioms.
You can always make up your own space with your own axioms, but you will have to prove your own theorems.
A: Here's an alternative definition of "topological space".
A topological space is a set $X$ together with a relation "___ is near ___" between points and subsets of $X$. The relation satisfies:


*

*No point is near the empty set

*If $P \in A$, then $P$ is near $A$

*$P$ is near $A \cup B$ if and only if $P$ is near $A$ or $P$ is near $B$

*If $P$ is near $A$ and every point of $A$ is near $B$, then $P$ is near $B$


That's it — a topological space is just a set of points together with a "nearness" relation that satisfies these axioms.
Then, an open set is simply a set whose points are not near its complement.
From one point of view, the purpose of topological space is to cut out all the extraneous information — many things can be stated and proven just in these terms, such as limits. Sometimes, this makes it easier to state and prove things. Other times, it lets us generalize.
For example, consider the extended real numbers. Its underlying set $\bar{\mathbb{R}}$ and consists of the real numbers $\mathbb{R}$ along with two extra points, which we will call $+\infty$ and $-\infty$.
One basis for the topology of $\bar{\mathbb{R}}$ is intervals of the form


*

*$(a,b)$ for (finite) real numers $a$ and $b$

*$(a, +\infty]$ for (finite) real numbers $a$

*$[-\infty, b)$ for (finite) real numbers $b$


You remember all of the different versions of "limit" you learned in introductory calculus, such as
$$ \lim_{x \to +\infty} \frac{x^2 + 1}{x + 2} = +\infty $$
? It turns out all of these are the same definition of limit, but applied to the extended real numbers rather than the ordinary real numbers.
Now, you could achieve the same thing talking only about metric spaces, but it would require inventing a new metric (e.g. $d(x,y) = |\arctan(x) - \arctan(y)|$, where we define $\arctan(\pm \infty) = \pm \pi/2$) and then you'd have to prove things about how the new metric relates to the usual metric and stuff, so it would be complicated and potentially confusing.
Topological spaces can also be applied to settings where it's not clear how to define a metric, or even when you can't even apply the notion of metric space at all.
An important example is used in algebraic geometry, one aspect of which is about studying solutions to polynomial equations. One defines the Zariski topology on the plane $\mathbb{R}^2$ to be the topology generated by a basis of open sets given by inequations of the form $f(x,y) \neq 0$, where $f$ is a polynomial in two variables. 
In the Zariski topology, the set of points
$$\{ P \in \mathbb{R}^2 \mid \|P\| \neq 1 \}$$
is an open set (being the solution space to $x^2 + y^2 \neq 1$), however the set of points
$$\{ P \in \mathbb{R}^2 \mid \|P\| <1 \}$$
is neither open nor closed.
Nearness in the Zariski topology has nothing to do with distance; instead it has more to do with how you can extend solution sets to polynomial equations; e.g. the point $(2, 0)$ is near the line segment from $(0,0)$ to $(1,0)$, because every polynomial that vanishes on that line segment (e.g. $y$ or $3y + x^2 y$)  also vanishes at the point $(2,0)$.
A: As Hurkyl demonstrates, general topological spaces can be defined in several ways.
Another way is with the concept of a Derived Set. Wikipedia begins,

In mathematics, more specifically in point-set topology, the derived
  set of a subset $S$ of a topological space is the set of all limit
  points of S. It is usually denoted by ${\displaystyle S'}$.
The concept was first introduced by Georg Cantor in 1872 and he
  developed set theory in large part to study derived sets on the real
  line.

This is amazing. Perhaps we should back up a bit and ask what is the benefit of studying sets. For Georg Cantor the creation of point-set topology came as an inspirational 'package deal'. 
I personally always prefers the term point-set topology over general topology. It emphasizes that instead of examining the elements in a set, we can talk about 'points' in a 'space'. This abstract and precise way of making 'spatial arguments' is something that Euclid, who developed 'point' geometry thousands of years ago, would no doubt appreciate.
I wonder if mathematicians will be throwing a big 150-year anniversary party in 2022 to recognize the astounding achievements of this man...
A: Metric spaces generalize distance, not topological spaces.
There are spaces which are not metric spaces but on which topology can be defined. For example the "long line" $\bf{R}^2$ with the dictionary order topology. or the quotient space  $\bf{Z}^2$x$\bf{R}$ where the points (0,x) and (1,x) are equivalent if x<0. (1,0) and (0,0) are distinct points but any attempt to put a metric on the space would result in these two points having distance 0 a no-no.
Topological spaces are generalizations of spaces with continuous maps, and there are many spaces which are not metrizable. For example algebraic varieties, which surfaces defined by the intersection of the zeros of complex polynomials. There are surfaces which often contain degeneracies which prevent a metric. This would preclude the whole field of algebraic geometry.
The whole point is that if you have a continuous map from one space into another space, then topological properties ( such as Hausdorfness, connectedness, paracompactness, compactness etc) are inherited by the image.
