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mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space. It is named after Kazimierz Kuratowski. The statement
Kuratowski_embedding
Polish mathematician and logician (1896–1980)
Kuratowski embedding; Kuratowski's free set theorem; Kuratowski's intersection theorem; Knaster–Kuratowski fan; Kuratowski–Ulam theorem; Kuratowski convergence
Kazimierz_Kuratowski
Graph that can be embedded in the plane
the sphere) are surfaces of genus 0. See "graph embedding" for other related topics. Kazimierz Kuratowski provided a characterization of planar graphs in
Planar_graph
axioms Kuratowski convergence Kuratowski–Zorn lemma Kuratowski's closure-complement problem Kuratowski's intersection theorem Kuratowski embedding Kuratowski–Ulam
List of things named after Kazimierz Kuratowski
List_of_things_named_after_Kazimierz_Kuratowski
Notion in metric geometry
finite metric spaces. Kuratowski embedding, an embedding of any metric space into a Banach space defined similarly to the Kuratowski map Injective metric
Tight_span
Algorithmic problem of finding non-crossing drawings
algorithm may be a planar graph embedding, if the graph is planar, or an obstacle to planarity such as a Kuratowski subgraph if it is not. Planarity
Planarity_testing
On forbidden minors in planar graphs
Komplexe", Math. Ann., 114: 570–590, doi:10.1007/BF01594196, S2CID 123534907. Kuratowski, Kazimierz (1930), "Sur le problème des courbes gauches en topologie"
Wagner's_theorem
Mathematical space with a notion of distance
Euclidean norm, and the maximum norm, respectively. More generally, the Kuratowski embedding allows one to see any metric space as a subspace of a normed vector
Metric_space
Mathematical proposition equivalent to the axiom of choice
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for
Zorn's_lemma
Function in mathematical analysis
ω-continuous families; hence still ω-continuous. Incidentally, by the Kuratowski embedding any metric space is isometric to a subset of a normed space. Hence
Modulus_of_continuity
greedy planar embedding can be found by applying the Knaster–Kuratowski–Mazurkiewicz lemma to a weighted version of a straight-line embedding algorithm of
Greedy_embedding
Embedding a graph in 3D space with no cycles interlinked
graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into three-dimensional Euclidean space
Linkless_embedding
a Riemannian metric g, Gromov proceeds as follows. One exploits Kuratowski embedding. One imbeds X in the Banach space L ∞ ( X ) {\displaystyle L^{\infty
Filling_radius
Embedding of Cantor set in 3-dimensional Euclidean space
In mathematics, Antoine's necklace is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected
Antoine's_necklace
Mathematical puzzle of avoiding crossings
proofs of this impossibility are known, and form part of the proof of Kuratowski's theorem characterizing planar graphs by two forbidden subgraphs, one
Three_utilities_problem
Area of discrete mathematics
embedding (or imbedding) of a graph in surface and linkless embedding, graph minors, crossing number, map coloring, and voltage graph. The embedding of
Graph_theory
Graph property
Verdière invariant at most k + 3 {\displaystyle k+3} . For instance, the two Kuratowski graphs K 5 {\displaystyle K_{5}} and K 3 , 3 {\displaystyle K_{3,3}} can
Colin de Verdière graph invariant
Colin_de_Verdière_graph_invariant
Graph in which every two vertices are adjacent
of the forbidden minors for linkless embedding. In other words, and as Conway and Gordon proved, every embedding of K6 into three-dimensional space is
Complete_graph
Graphs that differ only by edge subdivision
other three). In fact, a graph homeomorphic to K5 or K3,3 is called a Kuratowski subgraph. A generalization, following from the Robertson–Seymour theorem
Homeomorphism_(graph_theory)
planar by a certifying algorithm that outputs either a planar embedding or a Kuratowski subgraph. The extended Euclidean algorithm for the greatest common
Certifying_algorithm
Set theory concept
measurable cardinals were introduced by Stefan Banach (1930). Banach & Kuratowski (1929) showed that the continuum hypothesis implies that c {\displaystyle
Measurable_cardinal
Software library
function, If the graph is planar, a combinatorial embedding is produced as a witness. If not, a Kuratowski subgraph is returned. These values can then be
Library of Efficient Data types and Algorithms
Library_of_Efficient_Data_types_and_Algorithms
Describing a family of graphs by excluding certain (sub)graphs
(induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings
Forbidden graph characterization
Forbidden_graph_characterization
List of concrete topologies and topological spaces
horned sphere − A particular embedding of a sphere into 3-dimensional Euclidean space. Antoine's necklace − A topological embedding of the Cantor set in 3-dimensional
List_of_topologies
Existence of antipodal pairs in covers of spheres
-dimensional space R n + 1 {\displaystyle \mathbb {R} ^{n+1}} . In this embedding, the pairs of unit vectors ( x , − x ) {\displaystyle (x,-x)} are antipodal
Lusternik–Schnirelmann theorem
Lusternik–Schnirelmann_theorem
Non-crossing graph with vertices on outer face
outerplanarity. A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For k > 1 a planar embedding is said to be k-outerplanar if
Outerplanar_graph
an elementary embedding extendible cardinal A cardinal κ is called extendible if for all η there is a nontrivial elementary embedding of Vκ+η into some
Glossary_of_set_theory
Algorithm for counting perfect matchings in planar graphs
by using the Tutte matrix for the adjacency matrix in the last step. Kuratowski's theorem states that a finite graph is planar if and only if it contains
FKT_algorithm
Graph theory concept
a planar cover have an embedding into the projective plane? More unsolved problems in mathematics If a graph H has an embedding into the projective plane
Planar_cover
Extension of a formal language by the epsilon operator
for a later edition of Bourbaki that combined this notation with the Kuratowski definition of ordered pairs, this number grows to approximately 2.4 × 1054
Epsilon_calculus
Universal property of metric spaces
isometric embedding of X {\displaystyle X} into Aim ( X ) {\displaystyle \operatorname {Aim} (X)} ; this is essentially a generalisation of the Kuratowski-Wojdysławski
Metric space aimed at its subspace
Metric_space_aimed_at_its_subspace
Planar graphs have straight drawings
straight-line combinatorially isomorphic re-embedding of G in which triangle abc is the outer face of the embedding. (Combinatorially isomorphic means that
Fáry's_theorem
Theorem relating graph minors and topological embeddings
graph structure theorem may be looked at as a vast generalization of the Kuratowski theorem. A version of this theorem proved by Wagner (1937) states that
Graph_structure_theorem
Important problem in lattice theory
distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem. We denote by Con A the congruence lattice of an algebra
Congruence_lattice_problem
British-Canadian codebreaker and mathematician (1917–2002)
Using this fact, Tutte developed an alternative proof to show that every Kuratowski graph is non-planar by showing that K5 and K3,3 each have three distinct
W._T._Tutte
Kövari–Sós–Turán theorem (graph theory) Kruskal–Katona theorem (combinatorics) Kuratowski's theorem (graph theory) Lambek–Moser theorem (combinatorics) MacMahon's
List_of_theorems
Distance between two metric-space subsets
J\colon N\to L} into some common metric space L. Wijsman convergence Kuratowski convergence Hemicontinuity Fréchet distance Hypertopology Rockafellar
Hausdorff_distance
Mathematical tree of cycles
Nordhaus, E. A.; Ringeisen, R. D.; Stewart, B. M.; White, A. T. (1972), "A Kuratowski-type theorem for the maximum genus of a graph", Journal of Combinatorial
Cactus_graph
Great circle with a characteristic length
ISBN 978-3-030-27194-7 Katz, Mikhail (1991), "On neighborhoods of the Kuratowski imbedding beyond the first extremum of the diameter functional", Polska
Metric_circle
Continuity (topology) Homeomorphism Local homeomorphism Open and closed maps Embedding Germ Basis Subbasis Open cover Locally finite space Covering space Atlas
List of general topology topics
List_of_general_topology_topics
Math concept
{\displaystyle X} in c {\displaystyle {\mathfrak {c}}} such that for each closed embedding X ↪ Y {\displaystyle X\hookrightarrow Y} into a space Y {\displaystyle
Absolute_neighborhood_retract
Mathematical result or axiom on order relations
The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma. The Hausdorff maximal principle states that, in any partially ordered
Hausdorff_maximal_principle
contains very strong results about isomorphic measurable spaces, see Kuratowski's theorem. However, (a) the latter theorem is very difficult to prove,
Schröder–Bernstein theorem for measurable spaces
Schröder–Bernstein_theorem_for_measurable_spaces
Subset of a preorder that contains all larger elements
from the power set of X {\displaystyle X} to itself, are examples of Kuratowski closure operators. As a result, the upper closure of a set is equal to
Upper_and_lower_sets
Family of 7 undirected graphs
#R7, doi:10.37236/1033. Sachs, Horst (1983), "On a spatial analogue of Kuratowski's Theorem on planar graphs – an open problem", in Horowiecki, M.; Kennedy
Petersen_family
About mathematical functions
definition of the ordered pair (a, b) as {{a,1}, {b, 2}}. A few years later Kuratowski (1921) offered a definition that has been widely used ever since, namely
History of the function concept
History_of_the_function_concept
. K 5 {\displaystyle K_{5}} and K 3 , 3 {\displaystyle K_{3,3}} (the Kuratowski graphs) are the excluded minors for planar graphs and μ ≤ 3 {\displaystyle
Heawood_family
Mathematical operator
and A, with the upper adjoint being the embedding of A into P. Furthermore, every lower adjoint of an embedding of some subset into P is a closure operator
Closure_operator
Theorem about a certain class of control-flow graphs
of cyclomatic complexity, Thomas J. McCabe described an analogue of Kuratowski's theorem for the control-flow graphs (CFG) of non-structured programs
Structured_program_theorem
Axiomatic set theory devised by W.V.O. Quine
{\displaystyle R_{\alpha }} . If ( x , y ) {\displaystyle (x,y)} is the usual Kuratowski ordered pair (two types higher than x {\displaystyle x} and y {\displaystyle
New_Foundations
Mathematical set formed from two given sets
sets A {\displaystyle A} and B {\displaystyle B} , with the typical Kuratowski's definition of a pair ( a , b ) {\displaystyle (a,b)} as { { a } , { a
Cartesian_product
epi f {\displaystyle \operatorname {epi} f} as sets, in the Painlevé–Kuratowski sense of set convergence. Here, epi f {\displaystyle \operatorname {epi}
Epi-convergence
Mathematical ordering of a partial order
that the theorem had previously been proven by Stefan Banach, Kazimierz Kuratowski, and Alfred Tarski, again using the axiom of choice, but that the proofs
Linear_extension
the closure of a set S is a point of closure of S. Closure operator See Kuratowski closure axioms. Coarser topology If X is a set, and if T1 and T2 are topologies
Glossary_of_general_topology
Subgraph with contracted edges
contraction of edges can increase the genus of the embedding; therefore, planar graphs and the graphs embeddable on any fixed surface form minor-closed families
Graph_minor
Axiom of set theory
Schoenflies in 1905. Abstract algebra Hahn embedding theorem: Every ordered abelian group G order-embeds as a subgroup of the additive group R Ω {\displaystyle
Axiom_of_choice
convert nondeterministic automaton to deterministic automaton. Tarski–Kuratowski algorithm: a non-deterministic algorithm which provides an upper bound
List_of_algorithms
Relationship between two functors abstracting many common constructions
indicate the presence of adjunctions, as corresponding monads (cf. the Kuratowski closure axioms) a very general comment of William Lawvere is that syntax
Adjoint_functors
Axiomatic set theories based on the principles of mathematical constructivism
Secondly, for a notion weaker than finite, to be finitely indexed (or Kuratowski-finite) shall mean that there is a surjection from a von Neumann natural
Constructive_set_theory
Concept in mathematical set theory
2^{X}} (denoted by cl ( A ) {\displaystyle {\mbox{cl}}(A)} ) is the usual Kuratowski closure of a set, introduced in § 4, p. 20, is defined by cl ( A ) = {
Near_sets
KURATOWSKI EMBEDDING
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KURATOWSKI EMBEDDING