definition of containing convergence of sequence of subsets of set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of sequence.
Target Context
- The reader will have a definition of containing convergence of sequence of subsets of 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:
\( J\): \(\subseteq \mathbb{N}\), such that \(J \neq \emptyset\)
\( S'\): \(\in \{\text{ the sets }\}\)
\( s\): \(\in \{\text{ the sequences }\}\), with \(Dom (s) = J\) and \(Ran (s) \subseteq Pow (S')\)
\(*lim_{ci} s\): \(\subseteq S'\)
//
Conditions:
\(\forall j \in J (s (j) \subseteq lim_{ci} s)\)
\(\land\)
(
\(\vert J \vert \in \mathbb{N} \setminus \{0\} \implies lim_{ci} s = s (J_{\vert J \vert})\)
\(\land\)
\(\vert J \vert = \infty \implies \forall p \in lim_{ci} s (\exists N \in \mathbb{N} (\forall n \in \mathbb{N} \setminus \{0\} \text{ such that } N \lt n (p \in s (J_n))))\)
)
//
2: Note
This definition is not saying that \(lim_{ci} s\) always exists but is saying that if such a \(lim_{ci} s\) exists, it is called "containing convergence of \(s\)".
For example, when \(J\) is finite, if \(s (j) \subseteq s (J_{\vert J \vert})\) does not hold for a \(j \in J\), \(lim_{ci} s\) does not exist.
A typical case that \(lim_{ci} s\) exists is that \(s\) is non-decreasing, which means that for each \(j \lt j'\), \(s (j) \subseteq s (j')\), because \(lim_{ci} s = \cup_{j \in J} s (j)\), because for each \(j \in J\), \(s (j) \subseteq \cup_{j \in J} s (j)\), and for each \(p \in \cup_{j \in J} s (j)\), \(p \in s (j)\) for a \(j = J_N \in J\), but for each \(N \lt n\), \(J_N \lt J_n\) and \(p \in s (J_N) \subseteq s (J_n)\).
When \(lim_{ci} s\) exists, it is unique, because if \(S\) is another containing convergence, for each \(p \in S\), \(p \in s (j)\) for a \(j \in J\), but \(p \in s (j) \subseteq lim_{ci} s\), so, \(S \subseteq lim_{ci} s\), and \(lim_{ci} s \subseteq S\), symmetrically, so, \(S = lim_{ci} s\).
In fact, "containing convergence" or "\(lim_{ci} s\)" is not what the author have seen in any literature: this definition is concocted because the author needed it.
Compare to the definition of contained convergence of sequence of subsets of set.
