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It can be proved that arbitrary union of open sets is open. Suppose $v$ is a family of open sets. Then $\bigcup_{G \in v}G = A$ is an open set.

Based on the above, I want to prove that an arbitrary intersection of closed sets is closed.

Attempted proof: by De Morgan's theorem:

$(\bigcup_{G \in v}G)^{c} = \bigcap_{G \in v}G^{c} = B$. $B$ is a closed set since it is the complement of open set $A$.

$G$ is an open set, so $G^{c}$ is a closed set. $B$ is an infinite union intersection of closed sets $G^{c}$.

Hence infinite intersection of closed sets is closed.

Is my proof correct?

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    Why does DeMorgan’s rule definitely work on infinitary constructions?2017-08-28

1 Answers 1

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This is true, and your reasoning is correct too.

  • 7
    This is a comment2017-07-13