Def/Topological space

 =Basis= Suppose that $(X,T)$ is a topological space. A defines::basis for the topology $T$ is a subset $B \subset T$, such that:
 * If $U \in T$, then there exists a subset $S \subset B$, such that $U = \bigcup S$.

In other words, $B$ is a basis of $T$, if every element of $T$ (i.e., open set in $X$) can be expressed as a union of elements of $B$.

If $B$ is a basis of $T$, then $B$ satisfies the following properties: $$x \in b_3 \subset b_1 \cap b_2.$$
 * If $b_1, b_2 \in B$, and $x \in b_1 \cap b_2$, then there exists $b_3 \in B$, such that


 * If $x \in X$, then there exists $b \in B$ such that $x \in b$.