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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 ?

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

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