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
Best Answer
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.