The intersection of domains of holomorphy is again a domain of holomorphy

The original problem is this:

{Ωj}j∈J{Ωj}j∈J\{\Omega_j\}_{j\in J} is a family of domains of holomorphy. Prove that int(⋂j∈JΩj)\mathrm{int}\left(\bigcap_{j\in J} \Omega_j \right) is a domain of holomorphy.

I used Cartan-Thullen\textbf{Cartan-Thullen} first, which implies that UU is a domain of holomorphy if and only if it is holomorphically convex.

Now I can prove the theorem if |J||J| is finite by mean of the Lemma which reads:

K⊂U⊂V⟹ˆKU⊂ˆKV.K\subset U\subset V \Longrightarrow\widehat{K}_U \subset \widehat{K}_V.

However, I cannot prove the theorem if |J||J| is not finite. Specifically, I have no idea what the interior of the domains means here. I want to prove that the convex hull does not touch the boundary, but when interior operator is given, many points goes away. Then how can I prove that those points are not in the convex hull?

Thanks in advance for your help!

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