Question

In: Advanced Math

Is my proof that empty set is open and R is open correct?

Is that empty set is open and R is open correct? Give details Explaination.

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Expert Solution

Claim: The empty set is open.

 

Proof. Assume that the empty set is closed. Then, there must be one point such that any point in its ball is not inside of the empty set. However, the empty set has no point at all and therefore no such point and ball exists. This is a contradiction.

 

Claim: R is open.

 

Proof. Assume R is closed. Then its complement, the empty set, is open. But, we know that the empty set is closed. This is a contradiction.

Yes both are open.

Nipun Maity answered 2 years ago
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