I'm caring about some questions of non-standard analysis. I have found the only book talking about Baire Category Theorem, which is the book of Siu-Ah Ng. But I think the proof in this book is not correct, while it says:
By $^*X$ being $^*$complete, there is a unique $b \in \bigcup_{N<M\in ^*\mathbb{N}} B_M$: Then $b\in B_N \subset \mu(a) $ and $b \in \bigcap_{N\in ^*\mathbb{N}} W_n$.
Where $B_N$ is a open ball which has been chosen. I wonder if we can find the unique point by intersecting those open sets, and if we can do one after one on hyperfinite natural numbers, since we can't do induction on hyperfinite natural numbers?