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Microsoft Visual Studio 2015 Professional [x86/x64]. Autodesk Architect 2013 [x86/x64]. Autodesk Architect 2013 [x86/x64]…Q:
Let $G$ be a countable set and $D\subseteq G$ an open set. Then $D\cap[G]^\omega$ is countable.
Let $G$ be a countable set and $D\subseteq G$ an open set. Show that
$D\cap[G]^\omega$ is countable.
I tried to solve this with Tychonoff theorem and then compactness theorems for closed and discrete subsets, but I don’t know how to do the intersection of these two sets. I hope there is a simple solution to this problem which I can’t get. Thanks for your help.
A:
I would just try to guess a proof and see if it works.
Your argument is perfectly valid, you’re almost done. What you want is $D^\omega\subseteq [G]^\omega$. This is equivalent to $D\subseteq [G]^c$, which can be proved by using a compactness argument (since $[G]^c$ is compact and $D$ is closed).
Then you apply the general property of $[G]^c$ to $D$.
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