Pseudometric Topology

A topology on a set  whose open sets are the unions of open balls

where  is a pseudometric on ,  is any point of , and .

There is a remarkable difference between a metric and a pseudometric topology. The former is always , whereas the latter is, in general, not even . In fact, a pseudometric allows  for some distinct points  and , and then every open ball containing  contains  and conversely, so that no open set can separate the two points.