What does general in general topology really refer to? We use the term all the time without thinking about its origin.
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When I use it I mean to distinguish the subject matter from "specific" applications of topology where there is more structure than just that specified by the axioms for a topological space. (For example, a metric space, or a manifold or Riemannian manifold, or an algebraic variety, or you name it...)
I would add that in general, investigating terms like these for what they "really refer to" is a little misleading, because it sort of presumes that these terms were consciously designed by somebody (or a group of people) with a specific purpose in mind, instead of arrived at over the years sort of by accident, with no precise meaning--- only a general, vague, descriptive intent--- behind them.
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them. It is distinct from other branches of topology in that the topological spaces may be very general, and do not have to be at all similar to manifolds.
Other main branches of topology are algebraic topology, geometric topology, and differential topology. As the name implies, general topology provides the common foundation for these areas.