Topology glossary

Clear, one-line definitions of the Topology terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

basis (topology)

A collection B of open sets such that every open set in the topology is a union of members of B.

boundary

For a set S in a topological space, the points belonging to the closure of both S and its complement; written ∂S.

See: Free-surface boundary conditions, Absorbing boundary conditions (ABC)

closed set

A set whose complement is open; equivalently, a set containing all its limit points.

See: Open Sets, Closed Sets, and Topological Spaces

compact

A topological space in which every open cover has a finite subcover; in ℝⁿ this is equivalent to closed and bounded.

euler characteristic

A topological invariant χ = V − E + F for a polyhedron, generalizing to any compact surface; the sphere has χ = 2, the torus χ = 0.

finer

A topology τ₁ is finer than τ₂ on the same set when τ₂ ⊆ τ₁; τ₁ has at least every open set that τ₂ does.

interior

For a set S in a topological space, the union of all open sets contained in S; the largest open subset of S, written int(S).

manifold

A topological space that locally looks like ℝⁿ, every point has a neighborhood homeomorphic to an open subset of ℝⁿ.

See: Manifolds

metric

A function d(x, y) on a set assigning a non-negative distance, satisfying identity, symmetry, and the triangle inequality.

See: Metric Spaces and Distances, Daily report & productivity metrics

metric space

A set X equipped with a metric d: X × X → ℝ; generates a topology whose open sets are unions of open balls.

See: Metric Spaces and Distances

open ball

In a metric space, the set B(p, r) = {x : d(x, p) < r} of points strictly within distance r of p.

open in r^n

A subset U ⊆ ℝⁿ is open when every point p ∈ U has some open ball B(p, r) ⊆ U.

open set

A subset U of a topological space such that every point of U has a neighborhood contained in U; complement of a closed set.

See: Open Sets, Closed Sets, and Topological Spaces

real projective plane

The space ℝP² of lines through the origin in ℝ³; equivalently, a sphere with antipodal points identified.

topological space

A set X with a collection τ of subsets (the open sets) containing ∅ and X, closed under arbitrary unions and finite intersections.

See: Open Sets, Closed Sets, and Topological Spaces

topology generated

The smallest topology containing a given collection of sets (a subbase); formed by taking finite intersections then arbitrary unions.