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 $$mx \in I_x \subseteq S $$.
Best Answer
Intuitively:
A neighbourhood of a point is a set that surrounds that point. That is, if you move a sufficiently small (but non-zero) amount away from that point, you won't leave the set.
An interior point of a set is a point that is surrounded by the set. Note that this is really the same relation, only the subject has changed.
An open set is one which surrounds all its points. That is, wherever you are in that set, a sufficiently small move will not get you out.