definition of convergence of net with directed index set
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of net with directed index set.
Target Context
- The reader will have a definition of convergence of net with directed index 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: Structured Description
Here is the rules of Structured Description.
Entities:
\( D\): \(\in \{\text{ the directed index sets }\}\)
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( N\): \(: D \to T\)
\(*p\): \(p \in T\)
//
Conditions:
\(\forall N_p \subseteq T, \in \{\text{ the neighborhoods of } p\}\)
(
\(\exists j_0 \in D (\forall j \in D, j_0 \leq j (N (j) \in N_p))\)
)
//
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 \(j_0 \leq j\)
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.
