Question

In: Advanced Math

Two sets are separated if the intersection of the closure of one of the sets with...

Two sets are separated if the intersection of the closure of one of the sets with the other is empty.

a) Prove that two closed and disjoint sets of some metric space are separate.

b) Prove that two open and disjoint sets of some metric space are separate.

Solutions

Expert Solution

Check image for complete answer.

You just need the definition of closed and open sets.

For closed set A , we have Closure of A equals A.

For open set A , we have for each x in A there exists open neighborhood N which contains x and is completely contained in A.

x is in closure of A *iff* for each open neighborhood N containing x we have N intersection A is empty.

Colby Messinger answered 1 year ago
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