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Reading a book this question comes to my mind: is it possible to define metric spaces using only topological concepts without the explicit definition of a metric?

For example, can we say that all Hausdorff spaces are metric spaces?

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It is possible to define the class of metrizable spaces in purely topological terms. Indeed, there are many so-called metrization theorems that characterize the class of metrizable spaces topologically; here is a bundle of four of them.

Theorem. The following are equivalent for a topological space $\langle X,\tau\rangle$:

  1. $X$ is $T_3$ and has a $\sigma$-locally finite base.
  2. $X$ is $T_3$ and has a $\sigma$-discrete base.
  3. $X$ has open covers $\mathscr{G}_n$ for $n\in\Bbb N$ such that

    • if $G_0,G_1\in\mathscr{G}_{n+1}$ for some $n\in\Bbb N$, and $G_0\cap G_1\ne\varnothing$, then there is a $G_2\in\mathscr{G}_n$ such that $G_0\cup G_1\subseteq G_2$, and
    • whenever $x\in U\in\tau$, there is an $m\in\Bbb N$ such that $\bigcup\{G\in\mathscr{G}_n:x\in G\}\subseteq U$ for each $n\ge m$.
  4. $X$ is a paracompact Moore space.

  5. $X$ is metrizable.

However, a metric space by definition comes equipped with a metric, and a metric space with more than one point always has more than one metric that generates the same topology. Thus, we don’t actually have a metric space until we specify the metric, even when we know that there is one that generates the desired topology. If a topological space $\langle X,\tau\rangle$ satisfies one of the four conditions above, there is a way to construct a metric on $X$ that generates the topology $\tau$, but there are many other metrics on $X$ that generate the same topology.

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  • 1
    $\begingroup$ @Masacroso: You’re welcome. $\endgroup$ – Brian M. Scott Nov 7 '16 at 18:49
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    $\begingroup$ +1 for nice emphasis on distinction between "space for which a metric is assigned" and "space for which a metric can be assigned". $\endgroup$ – Dave L. Renfro Nov 7 '16 at 19:03
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    $\begingroup$ I misunderstood the question :) I like your answer by the way. $\endgroup$ – user384138 Nov 7 '16 at 19:39
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    $\begingroup$ @OpenBall: I wondered if that might have been the case. Thanks! $\endgroup$ – Brian M. Scott Nov 7 '16 at 19:40
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    $\begingroup$ such a great answer!! $\endgroup$ – user139708 Nov 8 '16 at 4:59

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