Open set - Wikipedia, the free encyclopedia

Closed set - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Empty_set

In Euclidean spaces, a set is closed if the limit of every convergent sequence of points in A, is in A itself.A subset U of the Euclidean n-space Rn is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rn whose Euclidean distance from x is smaller than ε, y also belongs to U. Equivalently, U is open if every point in U has a neighbourhood contained in U.

What you need to do in the question:

A) Show that is both open and closed.

B) are both open and closed. Try doing this only for , the rest are pretty much the same.

C) and D) are both done with proofs by contradiction. Can you think how?