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Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous. If f : X → Y is a function between topological spaces whose graph is closed in X × Y and if Y is a compact space then f : X → Y is continuous.
Generalized Borel Graph Theorem [11] — Let : be a linear map between two locally convex Hausdorff spaces and . If X {\displaystyle X} is the inductive limit of an arbitrary family of Banach spaces, if Y {\displaystyle Y} is a K-analytic space, and if the graph of u {\displaystyle u} is closed in X × Y , {\displaystyle X\times Y,} then u ...
So, if the open mapping theorem holds for ; i.e., is an open mapping, then is continuous and then is continuous (as the composition of continuous maps). For example, the above argument applies if f {\displaystyle f} is a linear operator between Banach spaces with closed graph, or if f {\displaystyle f} is a map with closed graph between compact ...
Homeomorphism (graph theory) – Concept in graph theory (closely related to graph subdivision) Homotopy#Isotopy – Continuous deformation between two continuous functions; Mapping class group – Group of isotopy classes of a topological automorphism group; Poincaré conjecture – Theorem in geometric topology; Universal homeomorphism
For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph so that the graph always remains entirely outside the cone. The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way δ {\displaystyle \delta } depends on ε ...
Function : is graph continuous if for all there exists a function : such that ((),) is continuous at .. Dasgupta and Maskin named this property "graph continuity" because, if one plots a graph of a player's payoff as a function of his own strategy (keeping the other players' strategies fixed), then a graph-continuous payoff function will result in this graph changing continuously as one varies ...
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A 2-dimensional hole (a hole with a 1-dimensional boundary). A 2-dimensional hole (a hole with a 1-dimensional boundary) is a circle (S 1) in X, that cannot be shrunk continuously to a point in X. An example is shown on the figure at the right. The yellow region is the topological space X; it is a pentagon with a triangle removed.