155: Convergence of Net with Directed Index Set

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definition of convergence of net with directed index set

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of net with directed index set.

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Target Context

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

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$D$: $\in \{\text{ the directed index sets }\}$
$T$: $\in \{\text{ the topological spaces }\}$
$N$: $:D\to T$
$\ast p$: $p\in T$
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Conditions:
$\mathrm{\forall }{N}_p\subseteq T,\in \{\text{ the neighborhoods of }p\}$
(
$\mathrm{\exists }{j}_0\in D(\mathrm{\forall }j\in D,j_0\le j(N(j)\in N_p))$
)
//

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2: Natural Language Description

For any directed index set, $D$, any topological space, $T$, and any net with directed index set, $N:D\to T$, any point, $p\in T$, such that for any neighborhood, $N_p\subseteq T$, of $p$, there is an index, $j_0\in D$, such that $N(j)\in N_p$ for every $j\in D$ such that $j_0\le j$

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3: Note

Although the relation of a directed index 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 index set, because any sequence on any topological space is a net with directed index set.

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References

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