Geometry & Topology

An affine map preserves affine combinations and can be expressed, after choosing coordinates, as a linear map followed by a translation.

An affine space is a space of points modeled on a vector space, with well-defined displacements but no distinguished origin.

Analytic geometry studies geometric objects through coordinates and equations, translating relationships between shapes into algebraic problems.

Archimedes was an ancient Greek mathematician and inventor whose surviving works established major results in geometry, mechanics, and hydrostatics.

A basis is a collection of open sets whose arbitrary unions constitute every open set of a topological space.

The Cantor set is a compact, uncountable subset of the real line with zero length, formed by repeatedly removing middle thirds from an interval.

A closed set is a subset of a topological space whose complement is open; in metric spaces, it contains every limit of its convergent sequences.

The closure of a subset of a topological space is the smallest closed set containing it, equivalently the set of points whose every neighborhood meets that subset.

A compact space is a topological space in which every open cover contains a finite subcover, generalizing key properties of closed, bounded subsets of Euclidean space.

A conic section is a plane curve obtained by intersecting a cone with a plane, encompassing circles, ellipses, parabolas, and hyperbolas.

A connected space is a topological space that cannot be partitioned into two disjoint, nonempty open subsets.

A convex combination is a weighted sum of points with nonnegative real coefficients whose sum is one.

The convex hull of a set is the smallest convex set containing it, equivalently the set of all finite convex combinations of its points.

A convex set contains the entire line segment joining any two of its points, making it a fundamental object in geometry, analysis, and optimization.

A covariant derivative differentiates vector and tensor fields using a connection, making the result independent of the chosen coordinates or local frame.

A vector operation in oriented three-dimensional Euclidean space whose result is perpendicular to both inputs and has magnitude equal to their spanned parallelogram’s area.

A dense set is a subset whose closure is the entire ambient space, so that every nonempty open region contains one of its points.

A differential form is a smooth field of alternating multilinear maps used to define coordinate-independent differentiation and integration on manifolds.

Differential geometry studies smooth spaces and their geometric structures using calculus, including curvature, distances, and connections between local and global properties.

Euclid was an ancient Greek mathematician whose Elements organized geometry and number theory into an influential deductive system.

A thirteen-book ancient mathematical treatise organizing geometry, proportion, and number theory through definitions, postulates, and deductive proofs.

Euclidean distance measures the straight-line separation between two points, calculated as the square root of the sum of squared coordinate differences.

A finite-dimensional space whose inner product determines the familiar geometry of distances, angles, and perpendicularity.

Felix Klein was a German mathematician who unified geometries through transformation groups and helped shape mathematical research and education.

A first-countable space is a topological space in which every point has a countable neighborhood base, allowing closure and continuity to be characterized using sequences.

A geodesic generalizes a straight line to curved spaces, locally minimizing distance in Riemannian geometry and describing free motion in general relativity.

The branch of mathematics that studies shape, size, relative position, and the properties of space, including Euclidean and non-Euclidean geometries.

A half-space is a region on one side of a hyperplane, with the boundary either included or excluded.

A Hausdorff space is a topological space in which any two distinct points have disjoint open neighborhoods.

A homeomorphism is a continuous bijection with a continuous inverse, expressing equivalence between topological spaces.

Homogeneous coordinates represent points by nonzero tuples defined up to a common scale, enabling a unified treatment of affine geometry, infinity, and projection.

A hyperplane is a flat subspace of codimension one, generalizing a line in a plane and a plane in three-dimensional space.

An isometry is a distance-preserving map between metric spaces, expressing exact geometric equivalence when it is also surjective.

Kakeya conjecture: the claim that every set containing a unit line segment in each direction of n-dimensional space must have full Hausdorff dimension n. Proved for n = 1, 2 and 3; open for n ≥ 4.

The Levi-Civita connection is the unique torsion-free connection that preserves the metric on a Riemannian or pseudo-Riemannian manifold.

A Lie group is a group with a compatible smooth manifold structure, providing a mathematical framework for continuous symmetry and its infinitesimal generators.

Linear separability is the property that two classes of points can be placed on opposite sides of a single hyperplane.

A manifold is a space that locally resembles Euclidean space, providing a framework for studying shapes, coordinates, and geometric structures.

A metric tensor is a smoothly varying symmetric bilinear form that defines local geometric measurements on a manifold, including lengths, angles, and spacetime intervals.

A neighborhood of a point is a set containing an open set that contains the point, expressing proximity without requiring a distance.

A net is a family indexed by a directed set that generalizes sequences and characterizes convergence in arbitrary topological spaces.

Non-Euclidean geometry studies spaces that depart from Euclidean geometry, especially hyperbolic and elliptic spaces with different rules for parallel lines.

An open set is a subset belonging to a topology, generalizing the idea that every point has a surrounding region contained within the set.

Pi (π) is the ratio of a circle's circumference to its diameter, an irrational and transcendental constant that appears throughout mathematics, science, and computing.

A polyhedron is a three-dimensional geometric object with polygonal faces; in convex mathematics, the term also denotes an intersection of finitely many half-spaces.

The product topology equips a Cartesian product of topological spaces with the coarsest topology making every coordinate projection continuous.

Projective geometry studies geometric properties preserved by projective transformations, emphasizing incidence, duality, and cross-ratio rather than distance and angle.

Ancient Greek philosopher and mathematician (c. 570–c. 490 BCE) who founded the Pythagorean brotherhood; few details of his life are reliably known.

The Pythagorean theorem relates the side lengths of a right triangle and underlies Euclidean distance, coordinate geometry, and orthogonal decompositions.

A quotient space identifies equivalent points of a topological space and equips the resulting set with the finest topology making the canonical projection continuous.

The Riemann curvature tensor measures intrinsic curvature through the noncommutativity of covariant differentiation and governs parallel transport and geodesic deviation.

A separable space is a topological space containing a countable dense subset, allowing its points to be approximated by a countable collection.

The subspace topology equips a subset of a topological space with open sets obtained by intersecting it with the ambient space’s open sets.

A tangent space is the vector space of first-order directions at a point of a differentiable manifold.

A tensor field assigns a tensor to each point of a space, describing geometric or physical quantities independently of coordinates.

A topological space is a set equipped with open subsets satisfying axioms that define continuity, convergence, and other spatial properties without requiring a distance.

Topology studies properties of spaces preserved by homeomorphisms, using concepts such as continuity, connectedness, compactness, and algebraic invariants.

A metric or uniform space is totally bounded if it admits a finite covering at every prescribed scale of closeness.

The triangle inequality states that a direct distance or the size of a sum cannot exceed the corresponding sum of distances or sizes.

Trigonometric functions relate an angle to ratios of side lengths, and they extend to all real numbers as periodic functions used across mathematics, science, and engineering.