#topology
12 concepts
Concepts
Continuity (Topological)
A function f: X → Y between topological spaces is continuous if the preimage of every open set in Y is open in X.
Homeomorphism
A homeomorphism is a continuous bijection with a continuous inverse. Homeomorphic spaces are topologically identical.
Connectedness
A topological space is connected if it cannot be separated into two disjoint nonempty open sets.
Manifold
Topological space locally resembling Euclidean space. Generalization of surfaces
Planar Graph
A graph that can be drawn on a plane without edge crossings. Euler's formula V-E+F=2 holds, and it contains neither K₅ nor K₃,₃ as a subdivision.
Fundamental Group
A group classifying loops in a space up to homotopy equivalence. Encodes 1-dimensional hole structure of spaces.
Homology Group
Abelian groups measuring n-dimensional holes in a space. Defined as quotient of cycles without boundary by boundaries.
Cohomology Group
Dual notion of homology with product structure (cup product). Connected to differential forms and de Rham cohomology.
Homotopy Groups
Groups classifying maps from n-spheres to spaces up to homotopy. Abelian for n≥2.
Covering Space
A continuous surjection with evenly covered neighborhoods at each point. In bijection with subgroups of the fundamental group.
CW Complex
A topological space built by attaching cells dimension by dimension. Standard approach for handling spaces in algebraic topology.
Euler Characteristic
A topological invariant of spaces, alternating sum of Betti numbers. For polyhedra, computed as V-E+F.