My professor proved this result in class, but I don't understand the "simple" direction. He said that the above result is in another words proving that Cl(a) = intersection of all closed sets that contain A.(adsbygoogle = window.adsbygoogle || []).push({});

So he proved Cl(A) subset of intersection of all of the closed sets containing A. And intersection of closed sets containing A is a subset of Cl(A). However, I don't understand this last direction.

He simply says:

Cl(A) = int(A) U Bd(A); by definition.

=> Cl(A) is one of the closed sets containing A.

=> intersection of closed sets containing A is a subset of Cl(A).

That's what I just don't get. I understand that Cl(A) is closed and it contains A, but I just don't see how we can then say that the intersection of all closed sets containing A has Cl(A) in it. I know its going to be one of those I'll be intersecting all the other closed sets with, but that Cl(A) is actually INSIDE the intersection I just don't see.

I'm having a massive brain freeze so an explanation would be appreciated.

Thanks

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# Cl(A) smallest closed set containing A.

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