Q:

Difference between open set and nbd

Plz clearly explain me difference between neighborhood of set and open set because according to me both definition are same.

A:

A neighbourhood of a point $p$ in a metric space $X$ with metric $d$ is any set that contains a ball of positive radius $B_r(p) = \{x\in X: d(x,p) < r\}$, $r > 0$, around $p$.  
An open set is a set that is a neighbourhood of each of its members.  
The difference between the two concepts is that a neighbourhood of $p$ might not be a neighbourhood of some other point $q$ in the set.  For example, in $\mathbb R$ with the usual metric, the closed interval $[-1,1]$ is a neighbourhood of $0$, but it is not a neighbourhood of $1$, and therefore it is not an open set.

