However it is enough in this. We shall understand the reason gradually.
The story will begin in \mathbb{R}^n . In \mathbb{R}^n we may define
some kinds of a distance function d(x,y).
Let a,b,c be elements in \mathbb{R}^n .
A distance function d(a,b) has to satisfy following conditions;
(1) d(a,a)=0 and if d(a,b)=0 , then a=b .
(2) d(a,b)=d(b,a) .
(3) d(a,b)+d(b,c)\geq d(a,c) .
Most popular form of a distance function in \mathbb{R}^n is
d(a,b)=\sqrt{(b_1-a_1)^2+(b_2-a_2)^2+\cdots +(b_n-a_n)^2} .
We will usually write d(a,b)=|b-a| or \lVert b-a \rVert .
Using a distance function, a neighborhood B of a point a in \mathbb{R}^n is defined.
B_{\epsilon}(a)=\left\{x| d(a,x)<\epsilon \right\} \quad (\epsilon>0)
Let A be a subset of \mathbb{R}^n . For any point x\in A ,
if there is a \epsilon >0 such that B_{\epsilon}(x)\subset A , then A is a open set of \mathbb{R}^n .
A closed set is a complementary set of the open set in \mathbb{R}^n.
As you may know well, it is a standard definition of a open set.
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