Motivation of generalizing the theory of metric spaces to the theory of topological spaces

Why does one need topological spaces if one has metric spaces?

What is the motivation of the abstract theory of topological spaces?

For me, the theory of metric spaces is quite natural. But I do wonder why there is the need of generalizing the whole theory... For instance, what are examples of topological spaces that are not metric spaces that really show that the theory of topological spaces is useful. There should be a strong reason, pathological examples don't suffice.

Metrics are often irrelevant. Even when working with metric spaces, it is not uncommon to phrase an argument purely in the language of open sets - and I have not infrequently seen mathematicians write proofs relying heavily on a metric and a complicated analytical argument when a simpler topological proof would suffice. Essentially, a topological space is a weaker structure than a metric space with a lot of the same logic.

Metrics are sometimes unnatural. There is a lot of study with topology where one works in metrizable spaces, but where there's no clear candidate for which metric to use - and it doesn't matter because we only care about topology. A common example of this is that the extended reals $\mathbb R\cup \{-\infty,\infty\}$ is metrizable, being homeomorphic to $[0,1]$, but it's really inconvenient to actually use the metric because every metric greatly distorts the ends of the real line - it's almost universally easier to think about the space in terms of open sets, noting that "close to $\infty$" means "bigger than some value" and things like that - where there's still some clear metric-like ideas, but we don't have to warp the line to do it. A lot of spaces fall into this category: for instance, projective space, one point-compactifications, and cartesian products all tend to fall into this category. Similarly, in a simplicial or CW complex, there tends to be a possibility of defining a metric, but we really don't care about it because we're more interested in the combinatorial structure of the connections or the topological properties than any idea of distance.

Some important (categorical) constructions don't work with metrics. A broader reason that metrics are not often used is because there's not really a good category of metric spaces. There is no notion of, for instance, an initial topology or an infinite product space - but these are extremely important in functional analysis. For instance, the Banach-Alaoglu theorem is really critical in functional analysis, especially in combination with theorems about duals such as the Riesz representation theorem, but these deal with the weak-* topology, which is usually not metrizable - and they often reason about these topologies via Tynochoff's theorem which simply has no analog in the theory of metric spaces. These theorems relate to incredibly important spaces that might have some nice properties (like Hausdorff or compact), but also fail others (like metrizable or even first countable). There are also wonderful things such as the Stone-Cech compactification which have surprising universal properties - but lead to incredibly badly behaved spaces which really cannot fit into the theory of metric spaces.

Some useful topological spaces really aren't at all like metric spaces. Examples such as the Zariski topology or the order topology on a poset often greatly contrast with the usual intuition behind topology - and allow familiar topological reasoning on an unfamiliar object. These spaces, however, are often not compatible with the theory of metric spaces, so there is not convenient flow of ideas that way.

This isn't to say that metric spaces are not useful, but they are good at describing spaces in which distance is a notion we want to think about. They are not so good as a basis for thinking about shapes and spaces more generally, where we might intentionally ignore distances to allow us to think about deformations and such.

While a topology feels at first much more abstract than a metric, it is all you need to build many, but not all, basic concepts you see in metric spaces (limits, pointwise continuity, compactness, etc.).

Here are three examples where topologies occur in math without metrics being used.

1. In functional analysis, the weak-$^*$ topology on the dual space $V^*$ of a Banach space $V$ is not metrizable if $V$ is infinite-dimensional.

2. Harmonic analysis in many respects generalizes from Euclidean space to arbitrary locally compact abelian groups (Fourier transform, Poisson summation formula, etc.), and analysis on locally compact possibly non-abelian topological groups is actively studied. Many important topological groups do not come with a natural metric on them, even if the topology is metrizable. For example, the adele group of a number field or the absolute Galois group of a number field are both important topological groups in number theory (the first is locally compact and the second is compact, both being Hausdorff) and while these topologies are metrizable I think it's fair to say that one hardly ever thinks about these groups in terms of a metric. If $G_i$ is an arbitrary family of compact topological groups, the product space $\prod_i G_i$ is a compact group using the product topology, but arbitrary (think uncountable) products of metric spaces need not be metric spaces in a reasonable way. In case you question the importance of arbitrary product spaces, look up the proof of Alaoglu's theorem in functional analysis. It uses an uncountable product of compact spaces, topologized with the product topology.

3. In algebraic geometry, the Zariski topology is extremely important and it is not just non-metrizable, but it is not even Hausdorff.

Conceptually, one of the good reasons for looking at metric spaces purely topologically is that it shows you what does not really depend on the choice of a metric. This becomes especially clear, I think, when you want to build quotient spaces and product spaces out of metric spaces. (Tori and Klein bottles are naturally defined as quotient spaces.)

If $(X,d_X)$ is a metric space and there's an equivalence relation $\sim$ on it, is the quotient space $X/\sim$ metrizable in a reasonable way? All these quotient spaces are naturally topologized using the quotient topology, which is the weakest topology (fewest open sets) on $X/\sim$ that makes the projection map $X \rightarrow X/\sim$ continuous. Some of these topologies are not metrizable since they are not Hausdorff. I'm not even sure what a metric analogue of "weakest topology" would be.

If $(X,d_X)$ and $(Y,d_Y)$ are metric spaces, is $X \times Y$ a metric space in a reasonable way? Too much time in Euclidean space suggests the metric $d((x,y),(x',y')) = \sqrt{d_X(x,x')^2 + d_Y(y,y')^2}$, but that square root is kind of artificial. The metric $\max(d_X(x,x'),d_Y(y,y'))$ is arguably nicer, but even better is to avoid all the fussiness over how to choose a metric and directly define a topology on $X \times Y$ from that on $X$ and $Y$: the product topology on $X \times Y$ is the weakest topology that makes the projection maps $X \times Y \rightarrow X$ and $X \times Y \rightarrow Y$ continuous.

I mentioned at the start that some concepts in metric spaces are not really expressible in terms of topology alone. Some important examples are uniform continuity of a function, uniform convergence of a sequence, and completeness. Weil introduced an abstract setting for that, uniform spaces, which includes both metric spaces and topological groups as fundamental examples. Other metric-dependent concepts are Lipschitz continuity, contractions, boundedness, and completeness. For example, the metric spaces $\mathbf R$ and $(0,1)$ are homeomorphic but the first one is complete and unbounded as a metric space while the second one is incomplete and bounded as a metric space.

A topology on a set $X$ is enough information to describe convergent sequences in $X$. However, a topology on $X$ is not always determined by the sequences in $X$ that converge in that topology (together with the limits). See here. If you generalize sequences to nets then you can say that a topology on $X$ determines and is determined by the convergent nets in $X$. See here.

Here's a non-pathological example. In $\mathbb C^n$, say that a set $A$ is open if$$A^\complement=\{p\in\mathbb C^n\mid(\forall p\in S):f(p)=0\},$$for some set $S$ of polynomial functions from $\mathbb C^n$ into $\mathbb C$.

This is the Zariski topology, which is non-metrizable and essential for Algebraic Geometry.

The Zariski topology is a good example of a topological space which is not metrizable.