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!!