This is in reference to page 153 of the notes found here http://www.math.lsa.umich.edu/~hochster/614F10/614.pdf
In the second paragraph, the author is considering inverse limit of a family of subsets of a certain set where the defining maps of the inverse systems are inclusions. I am not sure why the inverse limit is the intersection of the subsets.
For e.g. consider a set $X$ and four subsets $X_1,X_2,X_3,X_4$ s.t. $X_1\subset X_2$ and $X_3\subseteq X_4$, but $X_2\cap X_4=\emptyset$. Now, by the general construction of the inverse limit in the category of sets, we look at the elements of $X_1\times X_2\times X_3\times X_4$ s.t. any element in the second coordinate is also in the first coordinate and any element in the fourth coordinate is also in the third. So, the inverse limit is not empty, but the intersection is empty. What am I missing here?