Thread: interior points and proof of subsets

1. interior points and proof of subsets

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.

2. Hello.

One way to show that a set is closed is to show that it contains all of its limit points. So, take $A^c$ as in the problem, and put $E=\bigcup \{x\}$ where $x$ is a noninterior point of $A^c$.
Let $y$ be a limit point of $E$. Then, there is a sequence of points from $E$ that converges to $y$. If we can show that $y$ is an element of $E$ (i.e. a noninterior point of $A^c$), then $E$ is closed and we are done. To prove this, you should think about what happens if $y$ is not an element of $E$; in other words, what happens when $y$ is in the interior of $A^c$?