Question

Provide an example

1) A nested sequence of closed, nonempty sets whose intersection is empty.
2) A set A that is not compact and an open set B such that A ∪ B is compact.

3) A set A that is not open, but removing one point from A produces an open set.

4) A set with infinitely many boundary points.

5) A closed set with exactly one boundary point

Answer 1

1) You should take Fn=[n,+∞) then the intersection is empty.

This is correct. Every finite intersection has at least one point, but the infinite intersection is empty. As n goes from 1 to infinity, each integer at the start cannot be in the intersection of all. With n=2, the number 1 cannot be in the infinite intersection. With n=3, the number 2 cannot be in the infinite intersection, and so on. Choose any integer m. Then there is an n greater than m such that m is not in the infinite intersection.

2)Take A=(0.2,0.7)U{0,1}
then A is not compect
and let B=(0,1)
then AUB= [0,1], is a compect set

3)Take A= (0,1]
Now if we take,
A-{1}=(0,1) is open

4)Let A={n-1/m; n and m from set of natural numbers and m>1}
then all n≥1 are boundary points.

5)A Closed set is by definition a set whose complement is an open set. Therefore every element of complement set is exterior point and elements of set are interior point.Hence there not exist such closed set with exactly one boundary point.