I have been trying to construct a sequence of lower semi-continuous functions that will give indicator function as the infimum. In particular, I need a sequence $(f_i)_{i\in I}$ of lsc functions such that $\inf f_n(x)=1$ for $x\in\mathbb{R}\setminus\mathbb{Q}$ and $\inf f_n(x)=0$ for $x\in\mathbb{Q}$.
I taught $f_n(x)=1/n$ when $x$ is rational and $f_n(x)=1$ otherwise. However, this function is not lsc at irrational numbers. Thanks for any help!
Let $(r_n)$ be an ennumeration of ratinal numbers and $f_n(x)=1$ if $x \neq r_n$, $f_n(x)=0$ if $x = r_n$. Then each $f_n$ is l.s.c. and $\inf_n f_n(x)=1_{\mathbb R \setminus \mathbb Q}(x)$ for all $x$. (I have used the fact that indicator function of any open set is l.s.c.).