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?


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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    $\begingroup$ @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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