# Density of a set (proof check)

1. Oct 29, 2009

### utleysthrow

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

Given that O is an open set in W, and A is dense in W, prove that $$A \cap O$$ is dense in $$O$$

2. Relevant equations

3. The attempt at a solution

Looking at what is given,
$$O \subseteq W$$ because O is an open set in W, and $$W\subseteq\overline{A}$$ because A is dense in W

Suppose an arbitrary $$x \in O$$ (so $$x \in \overline{O}$$)
So $$x \in W$$
Which means $$x \in \overline{A}$$

Thus $$x \in \overline{A} \cap \overline{O}$$ or more specifically, $$x \in \overline{A \cap O}$$

That proves that $$O \subseteq \overline{A \cap O}$$, which should be enough to show that $$A \cap O$$ is dense in $$O$$