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Definition: [4] We say that f has a closed graph in X × Y if the graph of f, Gr f, is a closed subset of X × Y when X × Y is endowed with the product topology. If S = X or if X is clear from context then we may omit writing "in X × Y " Note that we may define an open graph, a sequentially closed graph, and a sequentially open graph in ...
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. [1] [2] [3] That is, a function : is open if for any open set in , the image is open in . Likewise, a closed map is a function that maps closed sets to closed sets.
The usual proof of the closed graph theorem employs the open mapping theorem.It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see closed graph theorem § Relation to the open mapping theorem (this deduction is formal and does not use linearity; the linearity is needed to appeal to the open mapping theorem which relies on the linearity.)
Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs; Open mapping theorem (complex analysis) – Theorem that holomorphic functions on complex domains are open maps; Surjection of Fréchet spaces – Characterization of surjectivity; Ursescu theorem – Generalization of closed graph, open mapping ...
Closed graph theorem – Theorem relating continuity to graphs; Open set – Basic subset of a topological space; Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets; Open mapping theorem (functional analysis) – Condition for a linear operator to be open (also known as the Banach ...
The closed graph theorem is an important result in functional analysis that guarantees that a closed linear operator is continuous under certain conditions. The original result has been generalized many times. A well known version of the closed graph theorems is the following.
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A monotone property of graphs is a property that is closed under subgraphs: if G has a monotone property, then so must every subgraph of G. Compare hereditary (closed under induced subgraphs) or minor-closed (closed under minors). Moore graph A Moore graph is a regular graph for which the Moore bound is met exactly.