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.
