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Theorem relating continuity to graphs
mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions
Closed_graph_theorem
Theorems connecting continuity to closure of graphs
the closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states
Closed graph theorem (functional analysis)
Closed_graph_theorem_(functional_analysis)
Finiteness of sets of forbidden graph minors
graph theory, the Robertson–Seymour theorem (also called the graph minors theorem) states that the undirected graphs, partially ordered by the graph minor
Robertson–Seymour_theorem
Area of mathematics
major theorems which are sometimes called the four pillars of functional analysis: the Hahn–Banach theorem the open mapping theorem the closed graph theorem
Functional_analysis
Linear operator whose graph is closed
unbounded operator. The closed graph theorem says a linear operator f : X → Y {\displaystyle f:X\to Y} between Banach spaces is a closed operator if and only
Closed_linear_operator
Generalization of closed graph, open mapping, and uniform boundedness theorem
and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle
Ursescu_theorem
Property of functions in topology
function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional
Closed_graph_property
Borel graph theorem is generalization of the closed graph theorem that was proven by L. Schwartz. The Borel graph theorem shows that the closed graph theorem
Borel_graph_theorem
On graphs with given symmetry groups
Frucht's theorem is a result in algebraic graph theory, conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite
Frucht's_theorem
Graph that can be embedded in the plane
whether any minor-closed class of graphs is determined by a finite set of "forbidden minors". This is now the Robertson–Seymour theorem, proved in a long
Planar_graph
Condition for a linear operator to be open
redirect targets Closed graph theorem – Theorem relating continuity to graphs Closed graph theorem (functional analysis) – Theorems connecting continuity
Open mapping theorem (functional analysis)
Open_mapping_theorem_(functional_analysis)
(combinatorics) Graph structure theorem (graph theory) Grinberg's theorem (graph theory) Grötzsch's theorem (graph theory) Hajnal–Szemerédi theorem (graph theory)
List_of_theorems
Semicontinuity for set-valued functions
\Gamma } has open lower sections then it is lower hemicontinuous. Open Graph Theorem—If Γ : A → P ( R n ) {\displaystyle \Gamma :A\to P\left(\mathbb {R}
Hemicontinuity
Planar maps require at most four colors
a graph coloring of the planar graph of adjacencies between regions. In graph-theoretic terms, the theorem states that for a loopless planar graph G {\displaystyle
Four_color_theorem
Complement of an open subset
Similarly, the closed graph theorem characterizes continuity of certain linear operators between Banach spaces by the closedness of their graphs. In the study
Closed_set
Fixed-point theorem for set-valued functions
spaces) and φ is required to be closed-valued in the alternative statement of the Kakutani theorem, the Closed Graph Theorem implies that the two statements
Kakutani_fixed-point_theorem
On forbidden subgraphs in planar graphs
In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states
Kuratowski's_theorem
On topological spaces where the intersection of countably many dense open sets is dense
functional analysis, BCT1 can be used to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle. BCT1 also shows that
Baire_category_theorem
Trail in which only the first and last vertices are equal
complement of a graph hole. Chordless cycles may be used to characterize perfect graphs: by the strong perfect graph theorem, a graph is perfect if and
Cycle_(graph_theory)
Subgraph with contracted edges
The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the complete
Graph_minor
Space where open mapping and closed graph theorems hold
with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains
Webbed_space
Set of eigenvalues of a matrix
subset. Here, I {\displaystyle I} is the identity operator. By the closed graph theorem, λ {\displaystyle \lambda } is in the spectrum if and only if the
Spectrum (functional analysis)
Spectrum_(functional_analysis)
Theorem on boundedness of symmetric operators
Otto Toeplitz. This theorem can be viewed as an immediate corollary of the closed graph theorem, as self-adjoint operators are closed. Alternatively, it
Hellinger–Toeplitz_theorem
Path in a graph that visits each vertex exactly once
Ore's theorems basically state that a graph is Hamiltonian if it has enough edges. The Bondy–Chvátal theorem operates on the closure cl(G) of a graph G with
Hamiltonian_path
On forbidden minors in planar graphs
In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite
Wagner's_theorem
Area of discrete mathematics
straight-line graph. Any planar graph can be represented as a planar straight-line graph by Fáry's theorem. The planar straight-line graph is the special
Graph_theory
Robertson–Seymour theorem characterizes minor-closed families as having a finite set of forbidden minors. mixed A mixed graph is a graph that may include
Glossary_of_graph_theory
Mathematical method in functional analysis
Hahn–Banach theorem may sometimes be used to show that an extension exists. However, the extension may not be unique. Closed graph theorem (functional
Continuous_linear_extension
On coloring infinite graphs
In graph theory, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. It states that,
De Bruijn–Erdős theorem (graph theory)
De_Bruijn–Erdős_theorem_(graph_theory)
On converting relations to functions of several real variables
and the implicit function theorem gives analytic conditions under which there exists a function f {\displaystyle f} whose graph belongs to the given curve
Implicit_function_theorem
Mathematical theorem about Banach spaces
spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved
Closed_range_theorem
Methodic assignment of colors to elements of a graph
graph introduced by Shannon. The conjecture remained unresolved for 40 years, until it was established as the celebrated strong perfect graph theorem
Graph_coloring
Describing a family of graphs by excluding certain (sub)graphs
forbidden graphs, the complete graph K5 and the complete bipartite graph K3,3. For Kuratowski's theorem, the notion of containment is that of graph homeomorphism
Forbidden graph characterization
Forbidden_graph_characterization
Formula used in graph theory
In graph theory, a part of discrete mathematics, the BEST theorem gives a product formula for the number of Eulerian circuits in directed (oriented) graphs
BEST_theorem
Theorem in group theory
trees and for graphs of groups and Dicks' even more straightforward proof of Grushko's theorem (see, for example, ). Grushko's theorem is, in a sense
Grushko_theorem
Graph defined from a mathematical group
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract
Cayley_graph
Theorem in real analysis
the endpoints −r and r, as the graph of f has vertical tangents at those points. Since f(−r) = f(r), Rolle's theorem applies, and indeed, there is a
Rolle's_theorem
Cubic graph with 10 vertices and 15 edges
bridgeless graph has a cycle-continuous mapping to the Petersen graph. More unsolved problems in mathematics In the mathematical field of graph theory, the
Petersen_graph
On tangency patterns of circles
packing theorem applies to any polyhedral graph and its dual graph, and proves the existence of a primal–dual packing, circle packings for both graphs that
Circle_packing_theorem
Gluing graphs at complete subgraphs
removed. And in yet other contexts, such as the graph structure theorem for minor-closed families of simple graphs, it is natural to allow the set of removed
Clique-sum
Theorem in calculus
divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through a closed surface to
Divergence_theorem
Theorem in topology
mathematics, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every Jordan curve (a plane simple closed curve) divides the plane
Jordan_curve_theorem
linear operators on a given space are closed. The closed graph theorem asserts that an everywhere-defined closed operator on a complete domain is continuous
Discontinuous_linear_map
Theorem in mathematics
In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating
Mean_value_theorem
Continuous function on an interval takes on every value between its values at the ends
function values has no gap, and the graph can be drawn without lifting a pencil from the paper. The corollary Bolzano's theorem states that if a continuous function
Intermediate_value_theorem
Graph linking pairs of comparable elements in a partial order
is Dilworth's theorem; these facts, together with the perfect graph theorem can be used to prove Dilworth's theorem from Mirsky's theorem or vice versa
Comparability_graph
Concept in topology
continuous. Secondly, there is a version of the open mapping theorem or the closed graph theorem due to Kuratowski: a continuous surjective homomorphism of
Polish_space
Any real function on R admits a continuous restriction on a dense subset of R
the Blumberg theorem guarantees that even this function has some dense subset on which its restriction is continuous. Closed graph theorem (functional
Blumberg_theorem
Property of artificial neural networks
Weisfeiler–Leman graph isomorphism test. In 2020, a universal approximation theorem result was established by Brüel-Gabrielsson, showing that graph representation
Universal approximation theorem
Universal_approximation_theorem
Linear operator equal to its own adjoint
R_{\lambda }} is closed (because A {\displaystyle A} is), so is R λ − 1 . {\displaystyle R_{\lambda }^{-1}.} By closed graph theorem, R λ − 1 {\displaystyle
Self-adjoint_operator
another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spaces. Ultrabornological spaces were
Ultrabornological_space
Locally convex topological vector space that is also a complete metric space
functional analysis, like the open mapping theorem, the closed graph theorem, and the Banach–Steinhaus theorem, still hold. Recall that a seminorm ‖ ⋅ ‖
Fréchet_space
Idempotent linear transformation from a vector space to itself
the closed graph theorem. Suppose xn → x and Pxn → y. One needs to show that P x = y {\displaystyle Px=y} . Since U {\displaystyle U} is closed and {Pxn}
Projection_(linear_algebra)
Type of vector space in math
graph is closed. By the closed graph theorem, a closed operator defined on all of a Hilbert space is bounded; hence a genuinely unbounded closed operator
Hilbert_space
countable graph have an unfriendly partition into two parts? Vizing's conjecture on the domination number of cartesian products of graphs Walescki's theorem for
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Relationship between derivatives and integrals
first fundamental theorem may be interpreted as follows. Given a continuous function y = f ( x ) {\displaystyle y=f(x)} whose graph is plotted as a curve
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Group whose Cayley graph is an initially subamenable graph
In mathematics, a sofic group is a group whose Cayley graph is an initially subamenable graph, or equivalently a subgroup of an ultraproduct of finite-rank
Sofic_group
Topological space arising from a usual graph
space projecting to a graph is also a graph. Graph homology Topological graph theory Nielsen–Schreier theorem, whose standard proof makes use of this
Graph_(topology)
Function spaces generalizing finite-dimensional p norm spaces
the counting measure on any finite set. As a consequence of the closed graph theorem, the embedding is continuous, i.e., the identity operator is a bounded
Lp_space
Any planar graph can be subdivided by removing a few vertices
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split
Planar_separator_theorem
Theorem in topology
Brouwer's theorem are for continuous functions f {\displaystyle f} from a closed interval I {\displaystyle I} in the real numbers to itself or from a closed disk
Brouwer_fixed-point_theorem
Nigerian mathematician and senator (1937–2018)
doi:10.1007/BF01896945. ISSN 0001-5954. ——— (1968). "-spaces and the closed-graph theorem". Proceedings of the Edinburgh Mathematical Society. 16 (2). Cambridge
Sunday_Iyahen
Mathematical method
compact and convex. If graph(Φ) is closed, then for every ε > 0 there exists a continuous function f : X → Y with graph(f) ⊂ [graph(Φ)]ε. Here, [ S ] ε {\displaystyle
Selection_theorem
Trail in a graph that visits each edge once
Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two
Eulerian_path
On linear-time algorithms for graph logic
study of graph algorithms, Courcelle's theorem is the statement that every graph property definable in the monadic second-order logic of graphs can be decided
Courcelle's_theorem
Mathematical theorem
In mathematics, the transfinite recursion theorem says a function can be defined using a recursion over a well-ordered set; for example, N {\displaystyle
Transfinite_recursion_theorem
Axiom of set theory
metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem. On every infinite-dimensional topological vector space there
Axiom_of_choice
Every simple polygon with more than three vertices has at least two ears
In geometry, the two ears theorem states that every simple polygon with more than three vertices has at least two ears, vertices that can be removed from
Two_ears_theorem
Ideals in a Boolean algebra can be extended to prime ideals
leave out "Hausdorff" we get a theorem equivalent to the full axiom of choice. In graph theory, the de Bruijn–Erdős theorem is another equivalent to BPI
Boolean_prime_ideal_theorem
Linear operator on dense subset of its apparent domain
{\displaystyle T} might not be defined for all of X {\displaystyle X} . Closed Graph Theorem—If X , Y {\displaystyle X,Y} are Hausdorff and metrizable, T : D
Densely_defined_operator
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree
Taylor's_theorem
Mathematical propositions in network flow theory
In graph theory, approximate max-flow min-cut theorems concern the relationship between the maximum flow rate (max-flow) and the minimum cut (min-cut)
Approximate max-flow min-cut theorem
Approximate_max-flow_min-cut_theorem
Upper bound on a graph's Shannon capacity
In graph theory, the Lovász number of a graph is a real number that is an upper bound on the Shannon capacity of the graph. It is also known as Lovász
Lovász_number
Equivalence of distributive lattices and set families
family of sets that is closed under these operations, automatically form a distributive lattice, and Birkhoff's representation theorem states that (up to
Birkhoff's representation theorem
Birkhoff's_representation_theorem
Fixed-point theorem for smooth manifolds
transversality assumption for the graph of f and the diagonal should ensure that the fixed point set is zero-dimensional. Assuming M a closed manifold should ensure
Atiyah–Bott fixed-point theorem
Atiyah–Bott_fixed-point_theorem
Maximal subgraph whose vertices can reach each other
Numbers of components play a key role in Tutte's theorem on perfect matchings characterizing finite graphs that have perfect matchings and the associated
Component_(graph_theory)
Graph which partitions into a clique and independent set
perfect graphs from which all others can be formed in the proof by Chudnovsky et al. (2006) of the Strong Perfect Graph Theorem. If a graph is both a
Split_graph
Graph representing faces of another graph
mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each
Dual_graph
Vector space with a notion of nearness
hold in general for topological vector spaces: the closed graph theorem, the open mapping theorem, and the fact that the dual space of the space separates
Topological_vector_space
Concept in topology
space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces
Baire_space
number of unlabelled graphs with n {\displaystyle n} vertices is still not known in a closed-form solution, but as almost all graphs are asymmetric this
Graph_enumeration
Graph formed by touching unit circles
penny graph is a unit disk graph and a matchstick graph. Like planar graphs more generally, they obey the four color theorem, but this theorem is easier
Penny_graph
Mathematical graph relating to chess
In graph theory, a knight's graph, or a knight's tour graph, is a graph that represents all legal moves of the knight chess piece on a chessboard. Each
Knight's_graph
Map that satisfies a condition similar to that of being an open map
redirect targets Closed graph theorem – Theorem relating continuity to graphs Open set – Basic subset of a topological space Open and closed maps – Functions
Almost_open_map
Embedding a graph in a topological space, often Euclidean
In topological graph theory, an embedding (also spelled imbedding) of a graph G {\displaystyle G} on a surface Σ {\displaystyle \Sigma } is a representation
Graph_embedding
of graph theory, the sphericity of a graph is a graph invariant defined to be the smallest dimension of Euclidean space required to realize the graph as
Sphericity_(graph_theory)
Operation combining two oriented knots
opposite colors. The Jordan curve theorem implies that there is exactly one such coloring. We construct a new plane graph whose vertices are the white faces
Knot_(mathematics)
Mathematical game on a topological space
Luzin sieves; invariant descriptive set theory; Suslin sets; the closed graph theorem; webbed spaces; MP-spaces; the axiom of choice; computable functions
Topological_game
discussion of the maximum-flow minimum-cut theorem. Cederbaum's theorem applies to a particular type of directed graph: G = (V, E). V {\displaystyle V} is
Cederbaum's maximum flow theorem
Cederbaum's_maximum_flow_theorem
Graph data structure
the e-graph according to some cost function, usually related to AST size or performance considerations. E-graphs are used in automated theorem proving
E-graph
Condition for a mathematical function to map some value to itself
the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional
Fixed-point_theorem
Normed vector space that is complete
The Closed Graph Theorem—Let T : X → Y {\displaystyle T:X\to Y} be a linear mapping between Banach spaces. The graph of T {\displaystyle T} is closed in
Banach_space
Mathematical formula involving a given set of operations
the Abel–Ruffini theorem states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms. A simple example
Closed-form_expression
Theorems generalizing the Brouwer fixed-point theorem
Kakutani fixed-point theorem: Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty
Fixed-point theorems in infinite-dimensional spaces
Fixed-point_theorems_in_infinite-dimensional_spaces
Type of topological vector space
F:X\to Y} is called closed if its graph is a closed subset of X × Y . {\displaystyle X\times Y.} Closed Graph Theorem—Every closed linear operator from
Barrelled_space
Undirected, connected, and acyclic graph
In graph theory, a tree is an undirected graph in which every pair of distinct vertices is connected by exactly one path, or equivalently, a connected
Tree_(graph_theory)
Structure-preserving correspondence between node-link graphs
In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a
Graph_homomorphism
Study of graphs defined by geometric means
Fáry's theorem states that any planar graph may be represented as a planar straight line graph. A triangulation is a planar straight line graph to which
Geometric_graph_theory
formulate theorems that apply simultaneously to all of these subclasses and contain specific results for each subclass as special cases. Although graph C*-algebras
Graph_C*-algebra
Graph representing intersections between given sets
intersection graph of unit disks in the plane. A circle graph is the intersection graph of a set of chords of a circle. The circle packing theorem states that
Intersection_graph
CLOSED GRAPH-THEOREM
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CLOSED GRAPH-THEOREM
CLOSED GRAPH-THEOREM
CLOSED GRAPH-THEOREM
CLOSED GRAPH-THEOREM
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CLOSED GRAPH-THEOREM