Why are topological spaces interesting to study?

In introductory real analysis, I dealt only with $\mathbb{R}^n$. Then I saw that limits can be defined in more abstract spaces than $\mathbb{R}^n$, namely the metric spaces. This abstraction seemed "natural"to me. Then, I knew the topological spaces. However, this time the abstraction did not seem natural/useful to me. Then one runs into problems of classifying spaces into normal/ first countable ... In my opinion, these resulted from the high level of abstraction adopted by studying topological spaces. When one uses a more general definition for a space, it is possible that the number of uninteresting objects increase. I guess this is what happened here, we use a very general definition for topological spaces, we get a lot of uninteresting spaces, then we go back and make classifications such as normal, Hausdorff,..

I was trying to justify to myself why are topological spaces are good to study. The best and only reason I can propose is that the category $Top$ is bicomplete.

Question 1: (Alternatives to $Top$)

If this is the only reason, can't there exist a "smaller" category such that it contains all metric spaces and is bicomplete ?

Question 2: (History of topological spaces)

I mentioned that the abstraction from metric spaces to topological spaces does not seem very natural to me. I suspect that historically, metric spaces were studied before topological spaces. If this is the case, I'd like to know what was the motivation/justification for this abstraction.

Question 3: (Applications of non-metric topology outside topology)

I mentioned earlier that "we get a lot of uninteresting spaces". Perhaps I am wrong (I hope I am wrong). I would value non-metric topological spaces more, if I see examples of theorems such that:

1) The theorems are in a branch of mathematics outside Topology

2) The theorems are proven with the aid of topology

3) The topological part about the proof of the theorem is about a non-metric space

Edit: ${}$ non-artificial instances of non-metric spaces appearing in other branches of math are valuable as well.

Thank you

The space $\mathbb{R}^\mathbb{R}$, which is the space of all functions from $\mathbb{R}$ to itself with the topology of pointwise convergence, is not a metric space (it is not even first countable). This kind of function space arises in many areas of math. The issue is that only countable products of metric spaces need to be metric, but function spaces like this are uncountable products.

This addresses only the third of your questions.

There has been a lot of interesting work lately by Martín Escardó, Paul Taylor, and others that interprets computability in terms of topology. It turns out, for example, that a predicate function $p:X\to\def\Bool{\mathbf{Bool}}\Bool$ is effectively computable if and only if it is continuous. But to make it work you cannot give $\Bool = \{\mathbf{True}, \mathbf{False}\}$ a metric topology. Rather, the correct topology is the Sierpiński topology in which $\{\mathbf{True}\}$ is open and $\{\mathbf{False}\}$ is not.

Other topological notions turn out to be important. For example, consider the function $\def\fa{\mathtt{forall}}\fa$, which takes a computable predicate $p: X\to\Bool$, and returns the truth of $$\forall x\in X. p(x).$$ ($\fa$ is a mapping from $\Bool^X\to\Bool$.) It transpires that $\fa$ is computable if and only if $X$ is topologically compact. This has some weird-seeming implications: $\fa$ is not guaranteed to terminate on the natural numbers $\Bbb N$, but it is computable and guaranteed to terminate on the Alexandroff compactification $\Bbb N\cup \{+\infty\}$. And similarly since the Cantor set of all sequences $\Bbb N\to \Bool$ is compact, $\fa$ can be effectively implemented to give a correct result for any predicate $p$ defined on sequences, even though the space of sequences is uncountable.

QUOTE:non-artificial instances of non-metric spaces appearing in other branches of math are valuable as well.

Let $U$ be a non-empty open subset of $\mathbb{R}^n$, $n \ge 1$. A test function on $U$ is a smooth function $U \rightarrow \mathbb{R}$ with compact support. Let $D(U)$ be the set of test functions on $U$. This is a vector space over $\mathbb{R}$. We can define a certain topology on $D(U)$ which makes $D(U)$ locally convex and complete. This topology is not metrizable. The dual space of $D(U)$ is called the space of distributions on $U$.

http://en.wikipedia.org/wiki/Distribution_(mathematics)

Most answers focus on the third question, so I'll try to say something about the first and the second. I think metric spaces (with non-expanding maps) are not suitable for topology. The reason is metric spaces do not glue well enough.

Let $A$ and $B$ be metric spaces, and $C$ be a subspace of both (that is, there are isometries from $C$ to subspaces of $A$ and $B$). Is there a metric space which glues $A$ and $B$ along $C$? That's quite natural thing to do. Topologists love to glue spaces together. What is a sphere? It is two hemispheres (that is, disks) glued along the equator. Manifolds are spaces that could be glued from pieces of $\mathbb{R}^n$ along open subsets. And so on.

Well, let's try to build such a metric space. Take a union of $A$ and $B$ as sets and identify points "from $C$" in both. Then you need to specify a metric. How to measure distance between a point $x$ from $A$ and a point $y$ from $B$? There was no way to estimate it before glueing. Probably the only reasonable thing to try is to set $d(x, y)$ to be infimum of $d(x, z) + d(z', y)$ where $z$ is a point of $A$ and $z'$ is a point of $B$ which glues to $z$.

That would be nice, but here is a problem: if $x$ and $x'$ are two points of $A$, it may become quicker to go from $x$ to some point of $B$, travel in $B$, and then return to $A$ to $x'$ than to go from $x$ to $x'$ directly. The language of travelling is not mathematically meaningful here, it just means that there could exist a point $y$ in $B$ such that $d(x, y) + d(y, x')$ is less than $d(x, x')$. That happens in almost all cases.

It could be fixed. Well, you need to acknowledge the possibility of "going to $B$ and returning to $A$" several times, so in the end your formula for distance between two points would look like this: for all sequences of points $s_1,\ldots, s_k$ evaluate $d(x, s_1) + d(s_1, s_2) + \ldots + d(s_k, y)$ and take the infimum (for all $k$ and for all sequences).

It's also possible the infimum will be zero. You have no choice but to quotient such pairs of points together. I believe after these manipulations you get a correct "glueing" (that is, pushout) in the category of metric spaces with non-expanding maps, so the category isn't absolutely unreasonable. Let's put the issue of zero infimum aside for now.

Now what's wrong with this construction? It contradicts my intuition about what space means. I expect that if I attach something to a space and then ignore the added part, I get the original space back. It doesn't happen here: there is no way to recover metric on $A$ from that infimum-of-sums formula. $A$ maps into "glued" space, and it's even a monomorphism, but $A$ is not a part of glued space.

If you want $A$ to be a part of glued space, you need to identify $A$ with its image. So the objects in the category would be "sets with many equivalent metrics on them". Then the morphisms also have to change: morphism from $X$ to $Y$ is a mapping of underlying sets which becomes non-expanding map for some choices of metrics on the $X$ and $Y$ equivalent to given ones. Translating this condition to the language of one chosen metric leads to the usual epsilon-delta definition of continuous maps.

As soon as you get continuous maps, topological spaces are not far ahead. Thinking in epsilon-delta terms is almost the same as thinking about open sets. The definition of a topological space is more or less the most general thing for which there is a notion of continuity. The modern definition is surely not the first one, but all previous were quite close.

It was not produced while studying metric spaces, but rather when trying to formalize glueing spaces together and quotienting by subspaces. At least that's my interpretation: historically there were a lot of theorems about spaces glued from polyhedrons, and later for CW-complexes, so the convenience of glueing is absolutely necessary for the definition of a space.

It seems nobody mentioned spaces of distributions. These are duals of function spaces and they are endowed with the weak-topology. In general this topology is not metrizable. This is a fundamental construction in the modern theory of PDEs and there are plenty of books with many results (Hormander, Gel'fand and Shilov etc.). See also this answer