Topology

A topological space is a structure consisting of a set and a collection of open sets (topology). It allows defining continuity, convergence, and connectedness.

A function f: X → Y between topological spaces is continuous if the preimage of every open set in Y is open in X.

A homeomorphism is a continuous bijection with a continuous inverse. Homeomorphic spaces are topologically identical.

A space is compact if every open cover has a finite subcover. In ℝⁿ, this is equivalent to being closed and bounded.

A topological space is connected if it cannot be separated into two disjoint nonempty open sets.

The Euler characteristic is a topological invariant, calculated for polyhedra as vertices - edges + faces.

A metric space is a set equipped with a distance function (metric) between points. It's a special case of topological spaces.

A manifold is a topological space locally resembling Euclidean space. Smooth manifolds additionally have differentiable structure.

The fundamental group π₁(X) consists of homotopy equivalence classes of loops (closed paths) starting and ending at a point in space X.

Homotopy is a continuous deformation between two continuous functions. Homotopy equivalent spaces have the 'same shape' topologically.

A group classifying loops in a space up to homotopy equivalence. Encodes 1-dimensional hole structure of spaces.

Abelian groups measuring n-dimensional holes in a space. Defined as quotient of cycles without boundary by boundaries.

Dual notion of homology with product structure (cup product). Connected to differential forms and de Rham cohomology.

Groups classifying maps from n-spheres to spaces up to homotopy. Abelian for n≥2.

A continuous surjection with evenly covered neighborhoods at each point. In bijection with subgroups of the fundamental group.

A sequence of group homomorphisms where image equals kernel at each step. Short and long exact sequences are important.

A topological space built by attaching cells dimension by dimension. Standard approach for handling spaces in algebraic topology.

A topological invariant of spaces, alternating sum of Betti numbers. For polyhedra, computed as V-E+F.