TITLE: What is a topological space good for?
QUESTION [51 upvotes]: 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 $\tau_1$?

REPLY [2 votes]: 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...