EDB — 0Q8

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

Let \(X\) be a metric space, and \(A⊆ X\). 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 \(E=D\).