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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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While I can't answer in general (heh), a course in general topology usually includes topics like separation, completion, and metrization. A course in algebraic topology focuses on homotopy and homology. –  yjj Dec 11 '10 at 8:36
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Interestingly, just this weekend I was looking over the 1929 paper On general topology and the relation of the properties of the class of all continuous functions to the properties of space by Edward W. Chittenden for something I was working on, and I began wondering if this paper may have had any influence on the phrase "general topology" coming into use. For example M. H. Stone, Garrett Birkhoff, and a few others started using "general topology" in titles of papers in the 1930s. –  Dave L. Renfro Aug 21 '12 at 20:42
    
I've always taken "general" topology as interchangeable with "point-set" topology. –  amWhy Nov 19 '12 at 23:54
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up vote 15 down vote accepted

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.

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Wikipedia says:

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.

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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.

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