Thread: prove that D is dense

1. prove that D is dense

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?

2. Re: prove that D is dense

Originally Posted by wopashui
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's the basic duality that needs to be noticed $\displaystyle D\cap U=\varnothing$ implies $\displaystyle D\subseteq (M-U)$ and $\displaystyle M-U$ is closed and not the full space (since $\displaystyle U\ne\varnothing$). Get it?

3. Re: prove that D is dense

Originally Posted by wopashui
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?