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Are there any "realistic" examples of topological spaces that are not metric spaces. You are free to invent your own definition of "realistic". But, at a minimum, a realistic example is one that occurs naturally as part of some other problem, not something that is fabricated to provide a counterexample or a topology homework exercise. One good example is the topological vector space of all functions $f:\mathbb{R} \to \mathbb{R}$ under pointwise convergence. Any others?

If such examples are difficult to find, maybe there are results in the opposite direction: are there results that say something like "every topology that satisfies conditions X, Y, Z can be generated from a metric"? One example: a compact Hausdorff space is metrizable if and only if it is second-countable. Any other theorems like that?

Edit
I found this page, which does a pretty good job of answering my second question. So, that just leaves the first one.

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@TomekKania -- thanks. The example I like best so far is the topological vector space of all functions $f:\mathbb{R}\rightarrow\mathbb{R}\ \ $ under pointwise convergence. –  bubba Aug 24 at 11:43
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Test function space, in the context of theory of distributions –  Jean-Claude Arbaut Aug 24 at 12:22

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Ordinal spaces occur naturally, and any uncountable ordinal space is not metrizable.

You can also talk about Moore spaces, Stone-Cech compactification of $\Bbb N$ (which is compact but has is too big to be metrizable), there are Zariski topologies which are often non-metrizable, and there are plenty of Cantor cubes which are naturally occurring in set theory.

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Zariski topologies usually are not even Hausdorff. I am also not sure how natural the Moore spaces are. –  Tomek Kania Aug 24 at 11:49
    
Yes, but not Hausdorff implies not metrizable. As for Moore spaces, it's a natural generalization of metric spaces, but maybe I'm just boasting my newfound knowledge that from the consistency of a strongly compact cardinal, it is consistent that all normal Moore spaces are metrizable. –  Asaf Karagila Aug 24 at 12:14
    
Thanks. I've never heard of any of these, so they don't occur "naturally" in my world. Nature is relative, evidently. –  bubba Aug 24 at 12:30
    
@bubba: Yes, nature is relative. For me large cardinals occur all the time, and adding a real number to the universe is something of an ordinary and natural process. For other people diagrams occur naturally, and things like morphisms and natural transformations occur naturally. –  Asaf Karagila Aug 24 at 12:33
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@AsafKaragila: This has nothing to do with "realistic". Reality is that which, when you stop believing in it, doesn't go away (Philip K. Dick). So it may seem that nature is relative, but it isn't. Nature does not need all mathematics fabricated by what mathematicians do. Mathematics is general, indeed: so general that much of it applies, if at all, only to mathematics (Preston C. Hammer). –  Han de Bruijn Sep 12 at 10:53

Many important quotient spaces are not metric, nor even Hausdorff. This happens quite commonly for orbit spaces of group actions, which are very important in geometric group theory, in dynamical systems, etc.

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