# To prove that $X_{\infty}$ is normal

If $$((X_n),(f_n))$$ is a inverse limit system, and for each $$n$$, $$X_n\neq {\emptyset}$$ is compact and Hausdorff then $$X_{\infty}$$ is normal.

I know there is a theory that says: If $$((X_n),(f_n))$$ an inverse limit system , and for each $$n$$, $$X_n\neq{\emptyset}$$ is compact and Hausdorff then $$X_{\infty}$$ is Hausdorff and compact. So by using another theory that every compact, Hausdorff space is normal , can it be solved?

• What is your definition of "reversed boundary system"? According to Google this post is the first time that phrase has ever been written on the internet, so I'm guessing you're translating from some other language... – Eric Wofsey Jan 17 at 21:16
• @EricWofsey I think "inverse limit" is meant here. – Henno Brandsma Jan 17 at 23:19