Question

Give an example of a sequence of open sets G1,G2,G3, ... whose intersection is neither open nor closed.

Answer 1

Explanation:

First, let us define what a set that is neither opened nor closed mean.

When we say a set of rational numbers is neither open nor closed. We are saying it isn't open because all the neighborhood of a rational number contains irrational numbers, and its complement isn't open either because all neighborhood of an irrational number contains rational numbers.

Now, a set that is neither open nor closed is a set that has an interval with one end open and one end closed, for example, [3, 4).

If we are to construct a sequence of open sets whose intersection is [3,4)

Then we say that:

Assume that G(n) = (-1/n, 1)

Then all G(n) contains [3,4).