# Open/closed set and interior point problem

complexnumber

## Homework Statement

Let $$(X,d)$$ be a metric space and let $$A \subseteq X$$. Denote the interior of $$A$$ by $$A^o$$.

## Homework Equations

Prove that if $$A$$ is open or closed, then $$(\partial A)^o = \varnothing$$. (Is this still true if $$A$$ is not open or closed?)

## The Attempt at a Solution

I don't even know what $$\partial A$$ means. Can anyone tell me what this represents?

complexnumber
That notation denotes the boundary of A.
http://en.wikipedia.org/wiki/Boundary_(topology)

\begin{align*} \partial A = \overline{A} \cap \overline{X - A} = \{ x \in X | \forall \varepsilon > 0 (O_\varepsilon(x) \cap A \ne \varnothing \wedge O_\varepsilon(x) \cap (X-A) \ne \varnothing \} \end{align*}

If $$A$$ is closed, then $$\partial A \subseteq A$$. However, $$\forall x \in \partial A \forall \varepsilon > 0 ( O_\varepsilon(x) \cap (X - A) \ne \varnothing)$$, hence $$\forall x \in \partial A \nexists \varepsilon > 0 (O_\varepsilon(x) \subseteq \partial A)$$. There are no interior points.

If $$A$$ is open, then $$\partial A \subseteq X - A$$. However, $$\forall x \in \partial A \forall \varepsilon > 0 ( O_\varepsilon(x) \cap A \ne \varnothing)$$, hence $$\forall x \in \partial A \nexists \varepsilon > 0 (O_\varepsilon(x) \subseteq \partial A)$$. There are no interior points.

Is this correct? Is it still true if $$A$$ is not open or closed? How can I prove that?

Is it still true if $$A$$ is not open or closed? How can I prove that?
Consider $A = \mathbb{Q}$, $X = \mathbb{R}$. What is the boundary of $\mathbb{Q}$?