TITLE: "Algebraic" topologies like the Zariski topology?
QUESTION [51 upvotes]: The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence.  The entire subject of Algebraic geometry is based on this simple fact.
Question: Are there other categories of algebraic objects that have interesting natural topologies that carry algebraic data like the Zariski topology on a ring (spectrum)?  If they exist, what are they and how are they used?

REPLY [6 votes]: carry algebraic data like the Zariski topology on a ring (spectrum)? If they exist, what are they and how are they used? 

In model theory they define and study the 'algebraic data like the Zariski topology' irrespectively of where these data come from. These data are called Zariski geometries, and e.g. admit some intersection theory, can be used to prove Chow's lemma, admit some classification results in dim 1 etc.  You may want to have a look on 
the recent book of Zariski geometries and references therein (or the actual book Zariski Geometries : Geometry from the Logician's Point of View, by Boris Zilber). 
Also, the book has some examples which sometimes need some work.