description/proof of that supremum of lower semicontinuous maps is lower semicontinuous
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of lower semicontinuous map from topological space into \(1\)-dimensional extended Euclidean topological space.
- The reader admits the proposition that for any linearly-ordered set and any subset, any element of the set is the supremum of the subset if and only if the element is equal to or larger than each element of the subset and for each element of the set smaller than the element, there is an element of the subset larger.
Target Context
- The reader will have a description and a proof of the proposition that the supremum of any set of lower semicontinuous maps is lower semicontinuous.
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:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(\overline{\mathbb{R}}\): \(= \text{ the extended Euclidean topological space }\)
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{f_j: T_1 \to \overline{\mathbb{R}} \vert j \in J, f_j \in \{\text{ the lower semicontinuous maps }\}\}\):
\(f\): \(: T_1 \to \overline{\mathbb{R}}\), \(= Sup (\{f_j \vert j \in J\})\), which means that for each \(t_1 \in T_1\), \(f (t_1) = Sup (\{f_j (t_1) \vert j \in J\})\)
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Statements:
\(f \in \{\text{ the lower semicontinuous maps }\}\)
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2: Proof
Whole Strategy: Step 1: for each \(t_1 \in T_1\) and each \(r \lt f (t_1)\), take an open neighborhood of \(t_1\), \(U_{t_1} \subseteq T_1\), such that \(r \lt f_j (U_{t_1})\), and see that \(r \lt f_j (U_{t_1}) \le f (U_{t_1})\).
Step 1:
Let \(t_1 \in T_1\) be any.
Let \(r \lt f (t_1)\) be any.
There is a \(j \in J\) such that \(r \lt f_j (t_1)\), by the proposition that for any linearly-ordered set and any subset, any element of the set is the supremum of the subset if and only if the element is equal to or larger than each element of the subset and for each element of the set smaller than the element, there is an element of the subset larger.
As \(f_j\) is lower semicontinuous, there is an open neighborhood of \(t_1\), \(U_{t_1} \subseteq T_1\), such that \(r \lt f_j (U_{t_1})\).
Then, \(r \lt f_j (U_{t_1}) \le f (U_{t_1})\), because \(f\) is the supremum.
So, \(f\) is lower semicontinuous.
