# Why is metric space a Hausdorff space but not a topological space?

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!!

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.
• @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); ... – Lee Mosher Mar 14 '17 at 21:52