Notes claim that a lower semi-continuous function attains its minimum on closed subsets of a Polish space

I am currently reading Touchettes notes: https://arxiv.org/abs/0804.0327 on large deviation results and in Appendix B the author states that (in page 81, below eq. B3) that:The lower semi-continuity of $I$ guarantees that this function achieves its minimum on any closed ses ( a lower semi-continuous function has closed level sets) where $I:\mathcal{X}\to[0,\infty]$ is a lower semi-continuous function on a Polish space. But isn't this claim false? Take $I(x) = x$, $\mathcal{X} = \mathbb{R}$ and then you have a continuous function on a Polish space, and for a subset $\mathbb{R}=C\subset\mathbb{R}$ you surely do not attain the minimum of $I$.

Edit: I just realized it after posting this question: I suppose that we take the topology of the affinely extended real numbers, where $I$'s classical divergence to $\infty$ does not matter.