Let A Q Give examples of sets that are neither open nor clo

Let A = Q. Give examples of sets that are neither open nor closed but are both relative to Q.

Solution

Q is neither open nor closed.

Since none of the points in Q were interior points thus Q is not open.

Its complement R\\Q shared the same property, so it is not open. Hence, Q is not closed.

Let A={1/n / nN}

A is not closed since 0 is limit point of A but 0A.

A is not open since every ball around any point contains a point in R\\A.

Let A={(-1)ⁿn/n+1 / nN}

A is not closed since 1 and -1 are limit points of A. However, neither 1 nor -1 is an element of A, so A does not contain all of its limit points.

A is not open.

Since 1/2 A. For any > 0, the set V(1/2) = (1/2 , 1/2 + ) contains infinitely many numbers (e.g. min{1/2 + /2, 0}) that are not elements of A since no number between 1/2 and 0 is in A.