Foundations

A metric supplies a distance satisfying three axioms.

A topology specifies which subsets are open.

Compactness reduces every open cover to a finite one.

Continuity preserves openness through inverse images.

Triangle-angle relations depend on the geometry.

A topological space is connected when it cannot be partitioned into two nonempty disjoint open sets.

A metric space is complete when every Cauchy sequence has a limit in that space.

A self-map T of a nonempty complete metric space with d(Tx,Ty)≤k d(x,y), 0≤k<1, has one fixed point.

In a metric space, K is compact exactly when every sequence in K has a subsequence converging to a point in K.

Uniform continuity requires one δ>0 for each ε>0 to control d_Y(f(p),f(q))<ε whenever d_X(p,q)<δ, for all p,q.

A space X is simply connected when it is path connected and every continuous loop based at a∈X deforms to the constant loop while keeping its endpoints at a.

A smooth n-manifold has locally Euclidean charts with smooth invertible transition maps, together with Hausdorff and second-countable topology.

A covector is a linear functional on the tangent space, and a tensor is multilinear on tangent/cotangent inputs.

Differential k-forms are smooth alternating covariant tensors.

On a compact oriented smooth n-manifold with boundary, integrating d-omega equals integrating omega on the induced oriented boundary, for a smooth (n−1)-form.

de Rham cohomology quotients closed k-forms by exact k-forms.

On a compact smooth Riemannian manifold without boundary, every de Rham class has a unique harmonic representative.

Over an algebraically closed field k, polynomial zero sets V(I) have coordinate ring k[x1,...,xn]/I(V(I)).

An affine scheme Spec(R) pairs the space of prime ideals with a structure sheaf.

A ring is reduced when it has no nonzero nilpotents.

The fundamental group pi1(X,x0) consists of based loops modulo homotopies fixing endpoints, with concatenation.

A chain complex has boundary maps with boundary²=0, and homology is cycles modulo boundaries.

For smooth complex algebraic varieties, analytic topology, algebraic coordinate structure and smooth de Rham theory offer complementary invariants.