TITLE: meaning of topology on a finite set
QUESTION [6 upvotes]: I have just started learning topology on my own and it's been quite intuitive and well-motivated while it was defined on the set of real numbers. However, in many books authors, for the sake of simplicity,  usually begin demonstrating a topology on some small finite sets such as $\{1,2,3\}$. For example, they say that $\{\{\}, \{1,2\}, \{1,2,3\}\}$ is a valid topology, since it satisfies three axioms. And i didnt get it. I cant grasp idea of topology on finite sets. Ideas of open sets, or open balls (when a metric defined) are clear on real numbers, but for me they lose the meaning on finite sets, however it should be vice versa.
That's why i have a question:

What does we obtain conceptually when we define a topology on a finite set? Sense of nearness, connectedness? May be by defining a topology $\{\{\}, \{1,2\}, \{1,2,3\}\},$ we show continuity within the $\{1,2\}$ and in a sense isolate $3$?

REPLY [8 votes]: I would like to add that there are topologies defined on finite sets that arise naturally in other mathematical constructions.
An example is a scheme from algebraic geometry. See here for basics on the spectrum of a ring.
In short terms, if $A$ is a commutative ring, then the set $\operatorname{Spec}(A)$ of all prime ideals of $A$ has a natural topology, called the Zariski topology. By definition, a subset $X \subseteq \operatorname{Spec}(A)$ is closed if there is an ideal $I$ of $A$ such that $X = \{P \in \operatorname{Spec}(A): I \subseteq P\}$.
Now consider for example the spectrum of a discrete valuation ring $A$ with maximal ideal $m$. The spectrum $\operatorname{Spec}(A)$ then consists of only two elements $\{0, m\}$.
The Zariski topology on $\operatorname{Spec}(A)$ has the following open sets: $\emptyset, \{0\}, \{0, m\}$.
This is a typical example in algebraic geometry, and one indeed has the notions of connectedness, continuity, etc. which have geometric interpretations.
For more information, a classical reference is the GTM book Algebraic Geometry by Robin Hartshorne.