EDB — 0Q8

[0Q8] Prerequisites:[0PD]↺↻,[0PP]↺↻, [0GJ]↺↻, [0P6]↺↻.Difficulty:*.

Let $X$ be a metric space, and $A\subseteq X$. We want to study the ”open-close” operation $\overline{(A^{\circ })}$ (which is the closure of the interior of $A$).

• Show a simple example where $\overline{(A^{\circ })}$ is not contained $A$.

• Then write a characterization of $\overline{(A^{\circ })}$ using sequences and balls.

• Use it to show that the ”open-close” operation is idempotent, that is, if $D=\overline{(A^{\circ })}$ and then $E=\overline{(D^{\circ })}$ then $E=D$.