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!!

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 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 always Hausdorff.