Lemke (unpublished, undated):
A topology, in mathematical terms, is A SET OF CRITERIA FOR ESTABLISHING DEGREES OF NEARNESS OR PROXIMITY AMONG THE MEMBERS OF SOME CATEGORY. It turns a 'collection' or set of objects into a space defined by the relations of those objects. Objects which are more alike by the criteria are represented in this space as being closer together; those which are less alike are further apart. There can be multiple criteria, which may be more or less independent of one another, so that two texts, for instance may be closer together in one dimension (say horizontal distance), but further apart in another (vertical distance). What is essential, obviously, is our choice of the criteria, the parameters, that define similarity and difference on each dimension.
