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According to this thread: Is this proof that all metric spaces are Hausdorff spaces correct? metric space is indeed a Hausdorff space.

And in this thread: Are all metric spaces topological spaces? It shows that metric spaces are not necessarily topological spaces.

But Hausdorff spaces are actually defined as topological spaces with more properties.Hausdorff_space_definition

So I am confused... If anyone can help me, thanks!!

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The short answer is yes, all metrics spaces are Hausdorff topological spaces. The slight technicality that the second link is trying to get at is that, according to the most strict definitions metrics spaces are not topological spaces.

A metric space is a set with a metric. A topological space is a set with a topology. A metric is a function and a topology is a collection of subsets so these are two different things.

However, the fact is that every metric $\textit{induces}$ a topology on the underlying set by letting the open balls form a basis. Once this is done then we can think of the metric space as coming with a natural underlying topological space structure, and when we do this the topological structure is $\textit{always}$ Hausdorff.

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  • $\begingroup$ There's even more. We can also express most of the metrical concepts in the terms of topology. The most important one being continous functions and "equal" (i.e. homeomorphic) spaces. Using higher level language: category of metric spaces is isomorphic to a full subcategory of topological spaces. On the other hand metric spaces should not be treated as a special case of topological spaces. Some concepts are hard to express in terms of topology, i.e. everything that is related to the metric, e.g. isometries, completness, geometry. $\endgroup$ – freakish Mar 14 '17 at 16:47
  • $\begingroup$ @freakish: The category theoretic statement should be explicit about the morphisms in the context of metric spaces. There are other categories of importance with metric spaces as objects but with more restrictive kinds of morphisms, for example: Lipschitz maps; $1$-Lipschitz maps (aka distance non-increasing maps); ... $\endgroup$ – Lee Mosher Mar 14 '17 at 21:52

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