What is the relation between Neighbourhood of a point,Interior point and open set? I want to know what is the relation between of Neighbourhood of a point,Interior point and open set?
Definition: A set $N \subset \mathbb{R}$ is called the $\textbf{neighbourhood}$ of a point a, if there  exists an interval I containing a and contained in N, i.e ,$$a\in I \subset N $$.
Definition:     A point x is an interior point of a set S if S is a nbd of x. In other words, x is an interior point of S if $\exists $ an open interval $(a,b)$ containing x and contained in S , i.e., $$x\in(a,b)\subseteq S$$.
Definition: A set S is said to be open if it is a nbd of each of its point, i.e, $x\in S$, there exists an open interval $I_x$ such that $$mx \in I_x \subseteq S $$.

It might help to look at a simple example.  Let $S = [0,1]$.  This is a neighbourhood of each point of $(0,1)$, but not a neighbourhood of $0$ or $1$.
Since it is not a neighbourhood of $0$ or $1$, and those points are in $S$, it is not an open set.  The interior of $S$ is $(0,1)$.
