Special open sets and epsilonization

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Explores interior points, boundaries, adherence, and compact sets, including definitions and examples.

Covers the approximation of functions in Sobolev spaces using smooth functions.

Explores the change of variables theorem and examples of polar coordinates.

Explores ramification theory, residual fields, and discriminant ideals in algebraic number theory.

Covers the definition of open balls, important subsets, interior points, open sets, closed sets, adhesion points, and border points.

Explores adhesion, convergence, closed sets, compact subsets, and examples of subsets in R^n.

Explores differentiability, partial derivatives, Schwarz's theorem, and function continuity in mathematical analysis.

Covers manifolds, topology, smooth maps, and tangent vectors in detail.

Explores ramification and structure of finite extensions of Qp, including unramified extensions and Galois properties.

Covers the recap of Galois theory and emphasizes the transitivity of Galois groups.