topological spaces 2

The standard definition of a topological space in the preceding post might have surprised you. Because there are no distance functions, no definitions of open sets, and so on.

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.