2022-10-23

155: Convergence of Net with Directed Indices Set

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A definition of convergence of net with directed indices set

Topics


About: topological space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of convergence of net with directed indices set.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Definition


For any directed set, \(S\), any topological space, \(T\), and any net with directed indices set, \(f: S \to T\), any point, \(p \in T\), such that for any neighborhood, \(N_p\), of \(p\), there is an index, \(i_0 \in S\), such that \(f (i) \in N_p\) for every \(i \in S\) such that \(i_0 \leq i\)


2: Note


Although the relation of a directed set may be partial, for any Hausdorff topological space, there can be only 1 convergence, as is proved in a proposition.

Convergence of sequence on topological space is a convergence of net with directed indices set, because any sequence on any topological space is a net with directed indices set.


References


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