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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:

  1. 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 $3$?
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    $\begingroup$ A finite topological space is just a finite pre-order , see here, e.g. $\endgroup$ – Henno Brandsma Feb 28 at 11:49
  • $\begingroup$ What is your definition of 'a topology'? Have you verified that your example satisfies the definition? And do you know that not every topological space is a metric space? $\endgroup$ – Servaes Feb 28 at 19:13
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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.

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The main interest of defining a topology $\tau$ on a finite set $F$ is a pedagogical one. Since there are one finitely many subsets, given $A\subset F$ there are only finitely many choices for the closure and for the interior of $A$. Besides, $\mathring A$ must be an element of $\tau$ (which is, again, a finite set), and the same thing applies to $F\setminus\overline A$. So, you can find $\mathring A$ and $\overline A$ by simply picking the right element from $\tau$, and this will help you to grasp the idea that $\mathring A$ is the largest open subset of $A$ and that $\overline A$ is the smallest closed set containing $A$.

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In addition to what others have already said (especially the point raised in one of the answers - that their use is mainly "pedagogical"), those finite examples are good for two more things:

  • An endless source of examples and counterexamples. Want to show that being a "Hausdorff" space is a thing? Here you go: your finite example is one that is not Hausdorff. If you have a sequence $a_n$ in your space converging to $1$, it will at the same time converge to $2$.

  • Some topological constructions already work on finite spaces, and may be the easiest to understand in that context. I am thinking of e.g. discrete topology (every subset is an open set) and indiscrete/trivial topology (the only open sets are empty set and the whole set). Note that the former is even metrizable - you can take the distance between any two distinct elements to be $1$ and this metric yields the discrete topology.

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My topology professor used to say that topology is the "study of continuous functions". It is clear to most that continuous functions mapping $\mathbb{R}\rightarrow\mathbb{R}$ are very useful. Later, this is generalized to metric spaces. As you say continuous functions can also be defined there. Topology goes even further than this. The idea is that one can define continuity, without necessarily having a metric defined over your space.

This is the case in your example: There is no metric over $\{1,2,3\}$, for which the sets $\{\},\{1,2\},\{1,2,3\}$ are the only open sets. Your question is justified: What's the point of this? This topology does not have any intuitively appealing properties. As José already pointed out, the point of this example is a pedagogical one.

As you study topology, you will learn definitions to classify topological spaces (connectedness, compactness, etc.). These help to describe the behavior of continuous functions, defined over those spaces. In my opinion, a good argument to study topology at this level of generality is that there are topological spaces with such useful properties, for which there is no metric (not metrizable). An example is the weak topology over an infinite-dimensional normed vector space.

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