# why characteristic function of closed sets are upper- semi continious?

Prove that characteristic function of closed sets are upper- semi continious

My attempt : i know the definition of upper- semi continious

i,e $$\chi_A ^{-1} (-\infty, a)$$ is open for every $$a \in \mathbb{R}$$ where $$A \subset X$$

we know that under a continuous function, the inverse image of a closed set is closed

Now my confusion is that if characteristic function is closed then $$\chi_A ^{-1} (-\infty, a]$$ is closed for every $$a \in \mathbb{R}$$ which is the definition of lower semicontinious

But our motive is to show characteristic function of closed sets are upper- semi continious

My doubts : why characteristic function of closed sets are upper- semi continious ?

• $\chi_A^{-1} (-\infty.a)=X$ if $a \geq 1$ and $A^{c}$ is $a <1$. – Kavi Rama Murthy Feb 23 at 6:03