Let be $\phi:X\rightarrow \mathbb{R}$ a lower semi-continuos function then $X = \cup_{n=1}^{\infty}\phi^{-1}(-n,\infty)$.
Why this?
Let be $\phi:X\rightarrow \mathbb{R}$ a lower semi-continuos function then $X = \cup_{n=1}^{\infty}\phi^{-1}(-n,\infty)$.
Why this?
This is true for any function from $X$ to $\Bbb R$; it has nothing to do with semi-continuity.
$X=\varphi^{-1}[\Bbb R]=\varphi^{-1}\left[\bigcup_{n\in\Bbb Z^+}(-n,\to)\right]=\bigcup_{n\in\Bbb Z^+}\varphi^{-1}\big[(-n,\to)\big]$
simply because $\Bbb R=\bigcup\limits_{n\in\Bbb Z^+}(-n,\to)$.