$(X,T)$ is compact iff every family of closed set ${F}_{i \in I}$ with non-empty finite intersections has an overall intersection that is not empty.

Question

I just learned that a topological space is compact if and only if for every family of closed sets $\{F_i\}_{i \in I}$ for which every finite subfamily has non-empty intersection, then the intersection of the whole family is non-empty. But what would be an example of family $\{F_i\}_{i \in I}$ in $(0,1)$ for which the finite intersection of the family are non-empty but the intersection of the whole family is empty?

Answer 1

For instance,$$\left\{\left(0,\frac1n\right]\,\middle|\,n\in\mathbb N\right\}.$$Clearly, if $F\subset\mathbb N$ is finite, then$$\bigcap_{n\in F}\left(0,\frac1n\right]=\left(0,\frac1{\max F}\right].$$But$$\bigcap_{n\in\mathbb N}\left(0,\frac1n\right]=\emptyset.$$