I've encountered the term Hausdorff space in an introductory book about Topology. I was thinking how a topological space can be non-Hausdorff because I believe every metric space must be Hausdorff and metric spaces are the only topological spaces that I'm familiar with. my argument is, take two distinct points of a topological space like p and q and choose two neighborhoods each containing one of the two points with the radius d(p,q)/2. Now I claim that these two neighborhoods must be disjoint. Suppose that they are not disjoint, therefore there must exists a z such that z lies in both neighborhoods. using the triangle inequality, I can say that d(p,q)<=d(p,z)+d(z,q). since z is in both neighborhoods we have: d(p,z)<d(p,q)/2 and d(z,q)<d(p,q)/2. using the triangle inequality again we obtain: d(p,q)<=d(p,z)+d(z,q)<d(p,q)/2 + d(p,q)/2 = d(p,q) which is a contradiction. therefore every metric space must be Hausdorff. is this a correct argument? Now, How can we find a non-Hausdorff space? such a space must be so interesting because we can't distinguish between two points intuitively. am I right? Is there any famous examples of a non-Hausdorff space that can be visualized or understood intuitively?