Let A be a subset of R^d. and
set E=A Union (non interior points of A complement).
a) prove E is a closed subset of R^d
b) With A and E as above, let F be a closed subset of R^d, and suppose that A is a subset of F. prove that E is a suset of F.
Let A be a subset of R^d. and
set E=A Union (non interior points of A complement).
a) prove E is a closed subset of R^d
b) With A and E as above, let F be a closed subset of R^d, and suppose that A is a subset of F. prove that E is a suset of F.
Hello.
I'd like to help you with the first part.
One way to show that a set is closed is to show that it contains all of its limit points. So, take as in the problem, and put where is a noninterior point of .
Let be a limit point of . Then, there is a sequence of points from that converges to . If we can show that is an element of (i.e. a noninterior point of ), then is closed and we are done. To prove this, you should think about what happens if is not an element of ; in other words, what happens when is in the interior of ?
Good luck.