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)?

In the setting of effective Polish spaces, you are describing the lightface $\Sigma^0_1$ pointclass. These are well-studied in effective descriptive set theory. The canonical reference on this subject is Moschovakis's Descriptive Set Theory, which is currently available online by the author.
