Q:

Passing from basis of topology to regular definition of open sets

The following is the definition of basis for a topology.
If X is a set, a basis for a topology on X is a collection B of subsets of X such that 
1)For each x $\in X$ there is atleast one basis element B containing X.
2)If x belongs to the intersection of two basis elements, then there is $B_3$ 
containing X such that $B_3$ $\subset B_1 \cap B_2$.
A subset U of x is said to be open in X if for each x $\in$ U, there is a basis element B $\in \beta$ such that x $\in B$ and B $\subset U$.
I was wondering how does this definition satisfies like regular open intervals 
and doesn't satisfy [a,b] ? I mean how come does it satisfy regular definition of open sets that we know of from analysis..?

A:

It is possible for $[a,b]$ to be open according to the definition open that you've given. There are many topolgies on $\Bbb R$. One in which intervals such as $[a,b]$ are open is the discrete topology.  In this topology, all subsets of $\Bbb R$ are open.

