The Shape of Space: Geometry and Topology
The geometry mathematicians actually study, read in order: open sets and what they buy you, the single theorem that swallows the integral theorems of calculus, curvature made into a number for curves and surfaces, and the punchline that there are exactly three classical geometries, told apart by the sign of one quantity.
You can state the topology axioms and say what continuity and compactness mean in terms of open sets, run the dictionary between differential forms and vector fields and read the generalized Stokes theorem as the common ancestor of the fundamental theorem, Green, Stokes, and divergence, extract curvature and torsion from a curve and Gaussian curvature from a surface, and classify any geometry as flat, spherical, or hyperbolic by measuring a single triangle.
The language of nearness
Three axioms about open sets are the whole apparatus: continuity becomes a statement about preimages, and Heine-Borel turns compactness in Euclidean space into closed and bounded.
A metric is one machine for generating a topology and a basis is another; seeing that the Euclidean and taxicab metrics induce the same open sets is what separates topology from geometry.
One theorem, four names
The determinant is a signed volume, and its multilinear alternating axioms are exactly the defining axioms of a form; the entire calculus of forms starts at the parallelepiped.
One operator does the work of gradient, curl, and divergence, and the identity that applying it twice gives zero is what the rest of the chapter leans on.
The dictionary between forms and vector fields, with the Hodge star as translator, explains why the cross product is a three-dimensional accident rather than a general tool.
Charts and atlases are how calculus escapes Euclidean space: the sphere, the torus, and the projective plane become places where forms can be integrated, once orientability is settled.
The integral of a derivative is a boundary value: the fundamental theorem, Green, classical Stokes, and divergence are four costumes on this one line, and the algebra-topology duality it exposes seeds de Rham cohomology.
Curvature, measured
Curvature begins as the reciprocal of the osculating circle's radius; signed curvature and the turning number are the plane-curve rehearsal for everything that follows.
Curvature says how a space curve bends and torsion says how it twists out of its plane of bending; the fundamental theorem says those two functions are the curve, up to rigid motion.
Two principal curvatures multiply into the Gaussian curvature and average into the mean curvature, and the Theorema Egregium says the first is intrinsic: a surface knows its own curvature without reference to the space around it.
Total curvature over a closed surface equals two pi times the Euler characteristic: integrated local geometry is decided by topology alone, the first great bridge between the two subjects of this path.
The three geometries
The parallel postulate is the hinge: keep it and triangle angles sum to two right angles, drop all parallels and you are on the sphere, where Girard's theorem prices a triangle's area by its angular excess.
Infinitely many parallels is a perfectly consistent geometry, and the Poincare disk makes it concrete: geodesics are arcs meeting the boundary at right angles, the rim sits at infinite distance, and a triangle's area is its angular defect.
The sign of the Gaussian curvature sorts geometry into three universes, and uniformization says the topology of a closed surface picks which one it lives in; this competency closes the loop the topology stage opened.