Recall that a nest is a family of sets which is linearly ordered by inclusion.
This problem is from kelley's "general topology" problem 5.H. the necessity follows from the finite intersection property(FIP) if X is compact.
I attempt to prove the sufficiency by the following steps:
- If each nest of closed non-void sets has a non-void intersection. Let A be a family of closed sets with the finite intersection property.
- We can get B as a maximal family of closed sets which contains A and has the finite intersection property by Hausdorff maximal principle.
- Let C be a maximal nest in B.
I want to prove that the intersection of members of B is non-void. But I feel it hard to accomplish the steps 3 and 4 which may lead to a complete proof.
I need some hints. Thanks!
Let $\mathcal{A}=\{X_\alpha:\alpha<\kappa\}$ be a family of nonempty closed sets which satisfy FIP. (Where $\kappa$ be a cardinal.) We will prove that $\bigcap \mathcal{A}$ is nonempty by transfinite induction for $\kappa$.
If $\kappa$ is finite, then $\bigcap{A}$ is nonempty trivially. If it holds for every cardinal less than $\kappa$, then we can check that $$Y_\alpha= \bigcap\{A_\xi:\xi<\alpha\}$$ is nonempty (by inductive hypothesis). Also, we can check that $\bigcap_{\alpha<\kappa}X_\alpha=\bigcap_{\alpha<\kappa}Y_\alpha$ and $\{Y_\alpha:\alpha<\kappa\}$ is a decreasing chains of nonempty closed sets so by hypothesis $\bigcap_{\alpha<\kappa}Y_\alpha\neq\varnothing$.