Topology

The study of continuity and of the structures that support it — open sets, limits, compactness, connectedness — without a metric being required first. The corpus holds it in one place: Part V of gallier-quaintance-math-deep, which teaches topology immediately before differential calculus and for that purpose.

That ordering is the whole content of the page for now. In this spoke's other analysis texts (trench-real-analysis, klazar-elementary-real-analysis, baggett-analysis-single-variable) the topological facts appear as lemmas about the real line, proved where needed. Gallier and Quaintance separate them out and prove them once in general, then define the derivative as a continuous linear map between normed spaces — the same move that makes the multivariable derivative an object of linear-algebra rather than a table of partial derivatives, and the same one that matrix-calculus takes further.

Topology is not one of the five subject families named in this spoke's domain header. It is here because a text that develops it from first principles arrived, which is the spoke's own boundary rule working. Whether it becomes a family of its own depends on a second source; the corpus has no dedicated topology text, nothing on algebraic topology, and the fractals detour (Part V, ch. 38) is the only place near a metric-geometry thread.

Related

euclidean-manifold · tensor-analysis · gallier-quaintance-math-deep · real-analysis · linear-algebra · matrix-calculus · mathematics · synthesis