open sets

In a metric space $\Omega$  we shall define open sets. Given sets $A,B\subset \Omega$  and
the distance function $d(x,y)$ .

We call a set $B(x,\delta)\in\Omega$  a open ball centered at a point $x$  which has the radius $\delta$  if, for any element $x\in\Omega$ and a real number $\delta>0$ , $B$  is a set $\left\{ y | d(x,y)<\delta \right\}$ .

Using a open ball, a set $A$ is called a neighbourhood of $x$  if $A$ has a subset $B(x,\delta)$ .

Open sets in $\Omega$  is the set in which all elements has a neighbourhood of the element
and which contains the neighbourhood.

Namely, a set $O$  is called open if, for any point $x\in O$ . there is a real number $\delta$ 
such that a set $\left\{ y | d(x,y)<\delta \right\}$ belongs to $O$ .

$R^n$  is open and the empty set $\phi$ is too. (After this, you will find that these two sets are also closed sets. )

Open sets are the base of topological space.