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?

 

[VeeEight]

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

 

[complexnumber]

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

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

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

If A is open, then \partial A \subseteq X - A. However, \forall<br /> x \in \partial A \forall \varepsilon &gt; 0 ( O_\varepsilon(x) \cap A<br /> \ne \varnothing), hence \forall x \in<br /> \partial A \nexists \varepsilon &gt; 0 (O_\varepsilon(x) \subseteq<br /> \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?

 

[jbunniii]

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}?