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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.

bubba
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3 Answers3

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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.

Asaf Karagila
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    Zariski topologies usually are not even Hausdorff. I am also not sure how natural the Moore spaces are. – Tomasz Kania Aug 24 '14 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 '14 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 '14 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 '14 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 '14 at 10:53
  • @Han: I completely lost you after the first sentence. Are you implying that all my examples are not realistic because they don't have to do with reality? Because then everything in mathematics is not realistic, and the little that you are left with is the fact that taking three apples and two pears will give you the same amount of fingers on your right hand (assuming you're a healthy individual in this aspect). If you're trying to make some joke and references, I didn't quite get it (perhaps because I didn't use the word "realistic" anywhere here, maybe on purpose, I don't know). – Asaf Karagila Sep 12 '14 at 11:10
  • @HandeBruijn How most of mathematics applies only to mathematics? Because I was sure mathematics is applied to physics, economics, chemistry, biology, computing, ... well, all the sciences and lot of more things. There is a lot of mathematics being used everywhere. – Integral Jul 04 '16 at 15:24
  • @Integral: I think the point of that quote, is that a lot of what mathematicians do have no direct applications to other parts of the human experience. For example, anything involving numbers larger than $10^{100^{1000^{10000}}}$, or the notion of "exactly half", or other things which do not exist in our physical perception of reality. And yet, the word "natural" has interpretation in mathematics (we've all came across "naturally arising structure" or whatever), which makes things somewhat relative from one mathematician to another, what constitutes as "natural" to begin with. – Asaf Karagila Jul 04 '16 at 15:33
  • @Integral: Take e.g. this reference: Hamel basis and additive functions. Then please answer the following question: what part of this reference is relevant for any application outside mathematics itself? My claim is that only the functions in Theorem 4 and 7 of the paper are relevant (the keywords are continuous and measurable). This would mean that we have only two lines of applicable mathematics out of six pages. I leave it to you to calculate the percentage. – Han de Bruijn Jul 04 '16 at 16:09
  • @HandeBruijn You should ask what part of this reference is relevant for any application outside mathematics itself right now* ?* Because all mathematics is potentially applicable. You can bring a lot of examples of "pure" mathematics without application, but this is temporary notion. The fact we don't know applications yet doesn't mean the subject is intrinsically non applicable. I want to remember you one of the most famous examples: cryptography. It relies on Number Theory, one of the subjects that G.H Hardy once was proud to be useless outside mathematics. – Integral Jul 04 '16 at 16:26
  • @Integral: Denied, but I'll give you the last word (Honestly, it would require a meeting or two between us trying to settle these matters) – Han de Bruijn Jul 04 '16 at 18:53
  • Honestly, you guys should take it somewhere else. Real life is fine, but anywhere but an answer of mine, is preferable. – Asaf Karagila Jul 04 '16 at 19:57
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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.

Lee Mosher
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The projection of the complex plane onto the real axis gives a simple, naturally occurring, example of a topological space that is not a metric space. Define d(z1, z2) = |Re(z1)-Re(z2)|. This is not a metric on the complex plane because d(z1, z2) = 0 does not imply z1=z2. But the open sets defined the obvious way does define a topology on C. Any non-trivial projection onto a metric space can give a similar example.

Mark
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