Prove that a subset D of a metric space M is dense in M iff $\displaystyle D \bigcap U$ nonempty for every nonempty open set $\displaystyle U \subset M$
so given$\displaystyle D \bigcap U$ nonempty we need to show that cl(D)=M, how does this work?
Prove that a subset D of a metric space M is dense in M iff $\displaystyle D \bigcap U$ nonempty for every nonempty open set $\displaystyle U \subset M$
so given$\displaystyle D \bigcap U$ nonempty we need to show that cl(D)=M, how does this work?
Here is a slightly different discussion of this.
The statement that $\displaystyle x$ is a contact point of $\displaystyle D$ means that $\displaystyle x\in D$ or $\displaystyle x$ is a limit point of $\displaystyle D$ [i.e. $\displaystyle x\in cl(D)]$.
So you are asked to show that each point of $\displaystyle M$ is a contact point of $\displaystyle D$.
Suppose that $\displaystyle t\in M$ but $\displaystyle t\notin cl(D)$. Then there is a ball $\displaystyle \mathcal{B}(t;\delta)$ that contains no point of $\displaystyle D$.
WHY? And why is that a contradiction?