Basis of a topology Does every topology have a unique basis? What I mean is if there is more than one topology that can be obtained from a given basis, which topology should I consider when they say the topology generated by the given basis?

No, there can be many basis for the same topology (but a basis generates a unique topology). 
For example in $\mathbb{R}^2$, for $r>0$, let 
$$B((x_0,y_0),r):=\{(x,y)\in \mathbb{R}^2:(x-x_0)^2+(y-y_0)^2<r^2\}$$
and
$$S((x_0,y_0),r):=\{(x,y)\in \mathbb{R}^2:|x-x_0|+|y-y_0|<r\}.$$
Then $\{B((x_0,y_0),r): (x_0,y_0)\in \mathbb{R}^2, r>0\}$ and $\{S((x_0,y_0),r): (x_0,y_0)\in \mathbb{R}^2, r>0\}$ are two different basis (actually they have no set in common) for the euclidean topology in $\mathbb{R}^2$.
P.S. One more example. The topology $\big\{\emptyset,\{x\},\{y\},\{x,y\}\big\}$ for the finite set $\{x,y\}$ has the following bases:


*

*$\big\{\{x\},\{y\}\big\}$

*$\big\{\emptyset,\{x\},\{y\}\big\}$ 

*$\big\{\{x\},\{y\},\{x,y\}\big\}$

*$\big\{\emptyset,\{x\},\{y\},\{x,y\}\big\}$

