Under-continuation

In Mathematical, a under-continuation (or a extracted continuation ) is a continuation obtained by taking only certain elements (an infinity) of a continuation starting. This operation is sometimes called extraction .

Formally, a continuation is an application defined on the unit $\ N$ of the natural whole . It classically is noted $(u_n) _ {N \ in \ NR}$ . A under-continuation or extracted continuation is the made up one of U by a increasing application strictly $\ varphi: \ NR \ rightarrow \ N$ .

She is thus written in the form $(u_ {\ varphi (N)}) _ {N \ in \ NR}$ . In this context, the application $\ varphi$ is called extractor .

Properties

Is $(u_n) _ {N \ in \ NR}$ a continuation of elements of a topological Espace X which converges towards $l$ , then any continuation extracted from $(u_n) _ {N \ in \ NR}$ converges towards $l$ .
the limiting of the under-continuations of a continuation $(u_n) _ {N \ in \ NR}$ of a topological space X are called the values of adherence of the continuation $(u_n) _ {N \ in \ NR}$ . It is a closed part of X .
Of all limited Continuation of realities, one can extract a convergent under-continuation (see the article on the Théorème of Bolzano-Weierstrass for more details).

Notes and references of the article

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