I was reading about topology and I came across this statement:
Every point $x$ in metric space $(X,d)$ has a neighborhood, which a neighborhood of $x$ (denoted $N(x)$) is defined as there exists $\delta >0$ such that the open ball $B_{\delta} (x) \subset N(x)$
The proof just says that take $N(x)= X$ as your neighborhood. Anybody can shed some light on how you can take $X$ be your neighborhood and ensure that every open ball around $x$ will lie completely in $X$?