Whats the difference between a closed set vs a set thats NOT open?

1. Jan 29, 2010

michonamona

Let f: D->R be continuous. If D is not open, then f(D) is not open.

Why can they not replace 'not open' with closed?

Thank you

M

2. Jan 29, 2010

blkqi

Not open does not imply closed. Most subsets of the real numbers are neither open nor closed (e.g. N, or [0,1), etc.). On the other hand, the set of all real numbers is both open and closed.

If a set A is not open then there exists a point x in A such that x is not an interior point of A. For instance, 0 is not interior to the interval [0,1), so its not open. Yet the interval is still "half-open", so it can't be closed.

Last edited: Jan 29, 2010
3. Feb 2, 2010