# Open/closed set and interior point problem

1. Mar 23, 2010

### complexnumber

1. The problem statement, all variables and given/known data

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

2. Relevant 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?)

3. The attempt at a solution

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

2. Mar 23, 2010

### VeeEight

3. Mar 24, 2010

### complexnumber

\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?

4. Mar 24, 2010

### jbunniii

Consider $A = \mathbb{Q}$, $X = \mathbb{R}$. What is the boundary of $\mathbb{Q}$?