definition of containing convergence of map from open interval into power set of set at boundary
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of map.
Target Context
- The reader will have a definition of containing convergence of map from open interval into power set of set at boundary.
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:
\( (r_1, r_2)\): \(\in \{\text{ the open intervals of } \mathbb{R}\}\), where \(r_1\) can be \(- \infty\) and \(r_2\) can be \(\infty\)
\( S'\): \(\in \{\text{ the sets }\}\)
\( f\): \(: (r_1, r_2) \to Pow (S')\)
\(*lim_{ci, r_1} f\): \(\subseteq S'\)
\(*lim_{ci, r_2} f\): \(\subseteq S'\)
//
Conditions:
(
\(\forall r \in (r_1, r_2) (f (r) \subseteq lim_{ci, r_1} f)\)
\(\land\)
\(\forall p \in lim_{ci, r_1} f (\exists {r_1}' \in (r_1, r_2) (\forall r \in (r_1, {r_1}') (p \in f (r))))\)
)
\(\land\)
(
\(\forall r \in (r_1, r_2) (f (r) \subseteq lim_{ci, r_2} f)\)
\(\land\)
\(\forall p \in lim_{ci, r_2} f (\exists {r_2}' \in (r_1, r_2) (\forall r \in ({r_2}', r_2) (p \in f (r))))\)
)
//
2: Note
This definition is not saying that \(lim_{ci, r_1} f\) or \(lim_{ci, r_2} f\) always exists but is saying that if such a \(lim_{ci, r_1} f\) or \(lim_{ci, r_2} f\) exists, it is called "containing convergence of \(f\) at \(r_1\) or \(r_2\)".
A typical case that \(lim_{ci, r_1} f\) exists is that \(f\) is non-increasing, which means that for each \(r \lt r'\), \(f (r') \subseteq f (r)\), because \(lim_{ci, r_1} f = \cup_{r \in (r_1, r_2)} f (r)\), because for each \(r \in (r_1, r_2)\), \(f (r) \subseteq \cup_{r \in (r_1, r_2)} f (r)\), and for each \(p \in \cup_{r \in (r_1, r_2)} f (r)\), \(p \in f (r)\) for an \(r \in (r_1, r_2)\), but for each \(r' \in (r_1, r)\), \(r' \lt r\) and \(p \in f (r) \subseteq f (r')\).
A typical case that \(lim_{ci, r_2} f\) exists is that \(f\) is non-decreasing, which means that for each \(r \lt r'\), \(f (r) \subseteq f (r')\), because \(lim_{ci, r_2} f = \cup_{r \in (r_1, r_2)} f (r)\), because for each \(r \in (r_1, r_2)\), \(f (r) \subseteq \cup_{r \in (r_1, r_2)} f (r)\), and for each \(p \in \cup_{r \in (r_1, r_2)} f (r)\), \(p \in f (r)\) for an \(r \in (r_1, r_2)\), but for each \(r' \in (r, r_2)\), \(r \lt r'\) and \(p \in f (r) \subseteq f (r')\).
When \(lim_{ci, r_1} f\) exists, it is unique, because if \(S\) is another containing convergence at \(r_1\), for each \(p \in S\), \(p \in f (r)\) for a \(r \in (r_1, r_2)\), but \(p \in f (r) \subseteq lim_{ci, r_1} f\), so, \(S \subseteq lim_{ci, r_1} f\), and \(lim_{ci, r_1} f \subseteq S\), symmetrically, so, \(S = lim_{ci, r_1} f\).
When \(lim_{ci, r_2} f\) exists, it is unique, because if \(S\) is another containing convergence at \(r_2\), for each \(p \in S\), \(p \in f (r)\) for a \(r \in (r_1, r_2)\), but \(p \in f (r) \subseteq lim_{ci, r_2} f\), so, \(S \subseteq lim_{ci, r_2} f\), and \(lim_{ci, r_2} f \subseteq S\), symmetrically, so, \(S = lim_{ci, r_2} f\).
In fact, "containing convergence" or "\(lim_{ci, r_j} f\)" 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 map from open interval into power set of set at boundary.
