some definitions related to open sets 3 (connected spaces)

Open sets are a most important and basic concept. We will also understand it by following explanations.

Giving a metric space $\Omega$  and subsets $A,B\subset \Omega$ .
If $A,B\ne \phi$ , $A\cap B=\phi$  and both $A$  and $B$ are open sets,
$S=A\cup B$  is not a connected space (or set).
$S$  is called a disconnected space (or set).

A connected set means we can not divide it two open sets which are not empty
and of which the intersection is empty. If $S$  is connected, $A$  or $B$  should be empty.

A trivial example of a connected space is a set which consists of a point and the empty set.
Namely, a point is the connected space.

As it is a obviously definite thing that the line of real numbers is connected,
we get to a following basic proposition.

A connected set on the real number line is a interval. (a set is not empty and the interval may be open or closed. )

some definitions related to open sets 2 (isolated and limit points)

Given a metric space $\Omega$ , a open set $A\subset \Omega$  and a member $a\in A$ .

We say a point $a$  is isolated, if, for any $\delta>0$ , a intersection of the open ball
$B(a,\delta )$  and $A$  is $a$ . It means there is not any point of $A$  except $a$ 
in a neighborhood of $a$ .

If any neighborhoods of $a$  has infinite points of $A$ , $a$ is called a accumulation point
or a limit point.

These will be understood easily. However, these are important definitions in the topology.
Please try to assume relations of various kinds of a set.

some definitions related to open sets

We shall introduce some definitions related to open sets. Here is a metric space $\Omega$  and $A,B\subset\Omega$ .

At first, the closed set means the complementary set of a open set. If $A$  is a open set,
$X=A^c$  is a closed set. Hence, $\mathbb{R}$  and the empty set $\phi$  are both closed sets.
Then $\mathbb{R}$  and the empty set $\phi$  become open sets and closed sets, too.
In a general metric space we have to accept sets which have these two properties at a same time.
There are sets which are not open and closed. You may remember half open intervals.

The interior $X$  of a set $A$  is the maximum open set of the set $A$ . Of course, if $A$  is a open set, $X=A$ . In other words, $X$  is the open set including all open sets which belong to $A$ . We often write just like $X=\cup\left\{ Y\subset A | Y \mbox{ is open}  \right\}$ .

The closure $X$  of a set $A$  is the minimum closed set of the set $A$ . If $A$  is a closed set, $X=A$ . Correctly, $X$  is the minimum closed set which includes the set $A$ . We also often write $X=\cap\left\{ Y\supset A | Y \mbox{ is closed}  \right\}$ .

The boundary $X$ of a set $A$  is the set whose elements are the closure minus the interior.
Namely, the intersections of the neighborhoods of any elements in the boundary $X$  and $A$  is not empty,  and the intersections of the neighborhood and the complementary of $A$  is not empty, too.

These must be the most familiar definitions to you. However, you have to note that these are only based upon the definition of open sets.

open sets 2 (the intesection is open)

We will give a general proof by which, given open sets $O_1,O_2$  in a metric space $\Omega$ ,
the intersection of $O_1$  and $O_2$  is open.

A intersection of a finite number of open sets is open. Namely,
If $O_1,O_2,\cdots , O_k\in\Omega$ , $O_1\cap O_2,\cap \cdots  \cap O_k\in\Omega$  is open.

If a element $e$  is in $\cap_{i=1}^k O_i$ , as $e\in O_i$ for all $i=1,\cdots ,k$ ,
there is a $\delta_i>0$  such that open ball $B(e, \delta_i)$  is in $O_i$ .
We choose $\delta=\min(\delta_1,\cdots\delta_k)$ .
Then, since $B(e, \delta)\subset B(e, \delta_i)$ for all $i=1,\cdots ,k$ ,
$B(e, \delta)\subset \cap_{i=1}^k O_i$ .

Thus, we obtain the result which we want. We should remember that a set $O$  is open
if and only if, for any element $e\in O$ , there is a $\delta>0$  such that the open ball $B(e,\delta)\subset O$ .