Lower semicontinuous functions: Difference between revisions

Revision as of 20:45, 26 October 2020

Let $X$ be a metric space (or more generally a topological space). A function $f:X\to \mathbb {R} \cup \{+\infty \}$ is lower semicontinuous if

\{x\in X:f(x)>a\}=f^{-1}\left((a,+\infty ]\right)

is open in $X$ for all $a\in \mathbb {R}$ .^[1]

@@ Line 1: / Line 1: @@
 Let <math> X </math> be a metric space (or more generally a topological space). A function <math> f : X \to \mathbb{R} \cup \{ +\infty \} </math> is '''lower semicontinuous''' if
 :<math> \{ x \in X : f(x) > a \} = f^{-1} \left( ( a , +\infty ] \right) </math>
-is open in <math> X </math> for all <math> a \in \mathbb{R} </math>.<ref name="Craig">Craig, Katy. ''MATH 201A HW 3''. UC Santa Barbara, Fall 2020.</ref>
+is open in <math> X </math> for all <math> a \in \mathbb{R} </math>.<ref name="Craig">Craig, Katy. ''MATH 201A HW 1''. UC Santa Barbara, Fall 2020.</ref>
 ==References==