definition of map from topological space into \(1\)-dimensional extended Euclidean topological space lower semicontinuous at point
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of extended Euclidean topological space.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of map from topological space into \(1\)-dimensional extended Euclidean topological space lower semicontinuous at point.
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:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( \overline{\mathbb{R}}\): \(= \text{ the extended Euclidean topological space }\)
\( t_1\): \(\in T_1\)
\(*f\): \(: T_1 \to \overline{\mathbb{R}}\)
//
Conditions:
\(\forall r \in \mathbb{R} \text{ such that } r \lt f (t_1) (\exists U_{t_1} \in \{\text{ the open neighborhoods of } t_1\} (r \lt f (U_{t_1})))\)
//
\(f\) is called "lower semicontinuous at \(t_1\)".
2: Note
When \(f (t_1) = - \infty\), there is no such \(r\), so, the condition is vacuously satisfied.
