Let F be a non-empty family of subcontinua of a continuum X such that for any finite subfamily $F_{1},F_{2},...,F_{n}$ in F there is $C\in F$ such that
$C \subset F_{1} \cap F_{2} \cap... \cap F_{n}$
I want to prove that
$\cap \{C: C \in F\}$ is a continuum.
Does anyone know how to do it ?
I was thinking of using the following theorem:
Theorem: If X is a continuum and $A_{1}, A_{2},...$ are nested subcontinuum $A_{n+1} \subset A_{n}, n \in \mathbf{N}$ then $A= A_{1}\cap A_{2} \cap...$ is a continuum of X.
But I don't know if this is enough.
Any help is appreciated.