A metric space is a set of points with a defined distance function, called a metric, that measures the distance between any two points in the set.
A metric space is Hausdorff if for any two distinct points in the space, there exists open sets that contain one point but not the other. In other words, every point in a Hausdorff space has a neighborhood that does not contain any other points.
Hausdorff spaces have important properties that make them useful in many areas of mathematics. For example, they allow for unique limits of sequences and continuous functions, and they are essential for the development of topological spaces.
The proof involves showing that the Hausdorff property holds for the standard metric on any given metric space. This can be done by considering the open balls around each point and showing that they do not intersect.
Yes, a non-Hausdorff metric space can exist, but it would not have many of the useful properties that Hausdorff spaces have. In fact, many common spaces, such as the real line with the lower limit topology, are not Hausdorff.