Search results
Results From The WOW.Com Content Network
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.
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 ...
The Borel graph theorem, proved by L. Schwartz, shows that the closed graph theorem is valid for linear maps defined on and valued in most spaces encountered in analysis. [10] Recall that a topological space is called a Polish space if it is a separable complete metrizable space and that a Souslin space is the continuous image of a Polish space ...
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 ...
An odd hole is a hole of odd length. An anti-hole is an induced subgraph of order four whose complement is a cycle; equivalently, it is a hole in the complement graph. This terminology is mainly used in the context of perfect graphs, which are characterized by the strong perfect graph theorem as
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
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.
According to the company, the center hole of a Cheez-It cracker is called the “docker hole.” The reason behind the docker hole is to release air and moisture as the crackers bake.