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I'm not quite sure what you're asking, but under one way to interpret the question, an answer is that the theory of locales is a well-developed alternative to the classical theory of topological spaces. A locale is essentially the algebraic structure of the opens of a topological space (which is more or less equivalent to its structure as a site), considered in isolation from any "points" that the space might have. Locale theory (or "pointless topology") looks a little different from classical point-set topology, but can be used for many of the same purposes, and has certain advantages (e.g. it tends to work much better when using constructive logic, such as the internal logic of a topos). Some books on locale theory include Johnstone's Stone spaces, Picado and Pultr's Frames and Locales, and Part C of Johnstone's Sketches of an Elephant.