A notion of topology for computability A topology on a space $X$ is defined as a subset of the power-set of X, that is closed under arbitrary unions, finite intersections and includes the empty set and the full space. 
Is anybody aware of a modification of the notion of topology where closure is only under finite intersection and recursively enumerable collections of open sets?
Example: 
Consider the set of natural numbers, and let the set of open sets be the set of recursively enumerable sets of the natural numbers. Now for any finite collection of r.e. sets their intersection is also r.e. Furthermore, for an r.e. collection of sets, their union is r.e. 
Has any work been done on a notion similar to this (if indeed that makes sense)?

The answer to "Has work been done on $X$?" is usually yes.  If you Google "computable topology" you will get a fair number of hits.
