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I've started studying topology, and the impression I get is that it's all about studying spaces without using a metric. So we have to talk about these open sets instead. So basically a topological space is a generalization of a metric space.

What I haven't been able to find, though, is any example of a space I would be interested in that doesn't have a metric! I'm not looking for general "applications of topology", I'm looking for a specific non-metric topological space that is interesting outside of topology, that doesn't have a metric. Preferably as simple as possible; i.e. ideally you should be able to just write it as a set comprehension. But any interesting spaces without metrics qualify.

Elliot Gorokhovsky
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  • You might find this MathOverflow question to be of interest. Similarly, this Math StackExchange question might help; in particular the link to $\pi$-base given there. Just so you know, the term for such spaces is "non-metrizable," and they are a topic of research interest in their own right. – Will R Oct 01 '16 at 00:04
  • @WillR Wow! How did I not find those?! Feel free to close as a duplicate. – Elliot Gorokhovsky Oct 01 '16 at 00:05
  • Linked is an example of a topological space that shows up in algebraic geometry that is not metrizable. https://en.wikipedia.org/wiki/Zariski_topology – Kaj Hansen Oct 01 '16 at 00:06
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    "So basically a topological space is a generalization of a metric space." I think many would prefer to think "a metric is a specific kind of topological space". – fleablood Oct 01 '16 at 00:12
  • The set of all real functions with the topology of pointwise convergence, in other words, $\mathbb R^\mathbb R$ with the (Tychonoff) product topology. – bof Oct 01 '16 at 00:16
  • The Stone-Čech compactification of $\mathbb N$ aka the space of all ultrafilters on $\mathbb N.$ – bof Oct 01 '16 at 00:18
  • Weak topologies... – copper.hat Oct 01 '16 at 01:02
  • Sierpinski space $S={0,1}$ with the topology $T={\emptyset, S,{0}}.$ – DanielWainfleet Oct 03 '16 at 04:27

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There are a lot. A few simple examples you can look up are the Sorgenfrey line on $\Bbb R$, the cofinite topology on any infinite set, and for a really easy example, the indiscrete topology on any set with more than one point.

How easy it is to prove these aren't metrizable depends on how much you've seen already, but for the latter two showing that the spaces aren't Hausdorff is probably the easiest way.

Alex Mathers
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    I was given cofinite topology as an example by my teacher; however, I can't see how it's useful anywhere. Do you know of any applications outside of topology? – Elliot Gorokhovsky Oct 01 '16 at 00:08
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    @RenéG I think if you spend a lot of times trying to find where things like this are "applied" you'll end up disappointed. The fact of the matter is that although general topology as a whole ends up finding applications elsewhere, people study topological spaces for the sake of it. Part of that is coming up with interesting (counter) examples, so you see things like the cofinite topology defined. – Alex Mathers Oct 01 '16 at 00:14