Question

Intersection of arbitray family of closed sets is closed but union of arbitray family of closed sets may not be closed.

Answer 1

Answer:

It can be proved that arbitrary union of open sets is open. Suppose v is a family of open sets. Then ⋃G∈vG=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:

(⋃G∈vG)c=⋂G∈vGc=B. B is a closed set since it is the complement of open set A.

G is an open set, so Gc is a closed set. B is an infinite union intersection of closed sets Gc.

Hence infinite intersection of closed sets is closed.

Is my proof correct

Step-by-step explanation:

Answer 2

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Refer to the attachment

Step-by-step explanation:

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