Topology

thumb|upright=1.5|A three-dimensional model of a Figure-eight knot (mathematics)|figure-eight knot. The figure-eight knot is a [[prime knot and has an Alexander–Briggs notation of 41.]]

thumb|right|A right-handed three-dimensional Cartesian coordinate system used to indicate positions in space|class=skin-invert-image

thumb|Sierpiński carpet|Sierpiński Carpet - Infinite perimeter and zero area thumb|Highly magnified area on the boundary of the Mandelbrot set thumb|The Mandelbrot set: its boundary is a fractal curve with [[Hausdorff dimension 2. (Note that the colored sections of the image are not actually part of the Mandelbrot Set, but rather they are based on how quickly the function that produces it diverges.)|200x200px]] thumb|Mandelbrot set with 12 encirclements

subset an ordered set that consists of all elements between two given endpoints

set equipped with a metric (distance function)

mathematical problem

in mathematics, space bounded by a sphere

sequence whose elements become arbitrarily close to each other

rational number sequence 𝐵ₖ such that (𝑚+1)∑𝑛ᵐ=(ᵐ⁺¹₀)𝐵₀𝑛ᵐ⁺¹−(ᵐ⁺¹₁)𝐵₁𝑛ᵐ+(ᵐ⁺¹₂)𝐵₂𝑛ᵐ¯¹−(ᵐ⁺¹₃)𝐵₃𝑛ᵐ¯²+⋯

impossible object

alt=From left to right: a point (marked 0), a line (marked 1), a triangle (marked 2), and a tetrahedron (marked 3).|thumb|The four simplexes that can be fully represented in 3D space.

metric space in which cauchy sequence converges to an element of the space

topological property

family of subsets of a set whose union equals the whole set

axioms in topology defining notions of "separation"

the closed interval of real numbers between 0 and 1

part of the domain of a mathematical function

inherited or induced topology

set that can be constructed from open or closed sets by repeatedly taking countable unions and intersections

material with insulating bulk but conductive boundary

geometric property of an object which cannot be mapped to its mirror image by rotations and translations alone

American award for geometry or topology research

in topology, a continuous function between two points

set of continuous functions from a topological space to the unit interval [0,1] such that for every point x, there is a neighborhood of x where a cofinite number of the functions are 0, and such that the sum of all the function values at x is 1

subset of a topological space that is closed and has no isolated points

quantum field theory whose correlation functions do not depend on the spacetime metric

type of relation for subsets of a topological space

topological space consisting of equivalence classes of points in another topological space

equivalence class of objects sharing local properties at a point in a topological space

in mathematical topology, a path whose initial point is equal to its terminal point

"one-point union" of a family of topological spaces

topology of topological vector spaces

three disjoint sets that share a common boundary

quotient space; operation of suspension creates a way of moving up in dimension (dimension n + 1)

connected open subset of a finite-dimensional vector space

lemma that every Sperner coloring of a triangulated simplex contains a properly colored simplex

continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace

mathematical idealization of the surface of a 3D body

topological space with a distinguished point

totally ordered set that shares certain properties of the real line

hypothetical fault-tolerant quantum computer based on topological condensed matter

topological construction

mathematics glossary

simplicial complex associated to an open cover of a topological space

frechet filter

natural topology induced by the Euclidean metric

mathematical method for optimizing material layout under given conditions

a stronger analog of limit point

a common construction in topology where one topological space is attached or "glued" onto another

binary operation between two pointed spaces: X ⨳ Y = (X × Y) / X ∨ Y

theorem that the monoid generated by taking closures and complements of a given set in a topological space has size at most 14, attained by e.g. the set (0, 1) ∪ (1, 2) ∪ {3} ∪ ([4, 5] ∩ ℚ) of reals

the space surrounding an object

metric space in which every two points are connected by a unique geodesic arc

concept in chemistry

proof in topology

award to highlight outstanding contributions by women in the field of topology and geometry

lemma that every open subset of the reals is a countable union of open intervals

Generalization of topological interior

mathematical subject