Thread: metric spaces (generalizing a result)

1. metric spaces (generalizing a result)

For the set $\displaystyle E=\{z\in{C}:|z|<1\}$ determine if $\displaystyle E$ is closed.

I know what I need to show, namely that the set is open because every limit point of the set is not in the set. I'm having trouble articulating this in the language of math. Here's my ideas:

Consider the complex $\displaystyle p$ with $\displaystyle |p|=1$.

It is clear that for every neighborhood $\displaystyle N_r(p)\text{ }\exists\text{ }{q}\neq{p}$ such that $\displaystyle q\in{E}$, but since $\displaystyle p\notin{E}$ $\displaystyle E$ is open.

The thing that I worry about is the "it is clear." How do I show in general that every neighborhood at p where |p|=1 contains points in E?

2. Re: metric spaces (generalizing a result) Originally Posted by VonNemo19 For the set $\displaystyle E=\{z\in{C}:|z|<1\}$ determine if $\displaystyle E$ is closed.
If $w\in\mathbb{C},~|w|=1,~\&~\delta>0$ then the neighborhood $\mathcal{N}=\mathscr{B}_{\delta}(w)$ must contain points in $E$ and points in $E^c$. Try $w=1$