topological spaces 3

If $A_1$  and $A_2$  are open sets in $\Omega$, then $(A_1\cap A_2)$  is a open set. We have already proved it in "open sets 2(the intersection is open)" generally.

If $(A_1\cap A_2)$  has no elements, or $A_1$  is equal to $A_2$ ,
then the proof would even be clear.

In the case which $(A_1\cup A_2\cup\cdots )$  is open, the proof would not also be required.

Therefore, the definition of a topological space essentially gives the collection of open sets in $\Omega$ .

Precisely the collection of open sets is called the topology,
and $\Omega$  is called the support or the set theoretic support.

For example, as $(\Omega, \phi)$  is the collection of open sets,formally $(\Omega, \phi)$  gives a topology.
Let $\left\{ (-\infty,0)\cup (0,\infty)\right\}$  be $\Omega$ . $(-\infty,0)$  and $(0,\infty)$  give a topology, too.

That is to say, on a $\Omega$ , various kinds of a toplogical space can be defined.

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.

topological spaces

We shall give the standard definition of a topological space.

For a set $\Omega$ , given $\mathcal{F}$ which is the family of subsets of $\Omega$ 
(it is a element of the power set of $\Omega$) .
$\Omega$  is a topological space, if following conditions are satisfied;

(1)$\Omega\in \mathcal{F}$ , and $\phi\in \mathcal{F}$ .
(2)If $A_1\in \mathcal{F}$ and $A_2\in \mathcal{F}$, then $A_1\cap A_2\in \mathcal{F}$ .
(3)If $\left\{A_i(i=1,2,\cdots )\right\}$  is a family of the elements of $\mathcal{F}$ 
(namely, all $A_i$ is the element of $\mathcal{F}$), then $\cup_{i}A_i\in \mathcal{F}$ .

We also say $A_i$  a open set of $\Omega$ . 

It is equivalent to the preceding definition(coffee break 8-2).
However, the definition of a open set is given additionally.