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?
 A: 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:


*

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

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

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

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

A: Let $X$ be a set. 
For any family of subsets $\mathcal A$ of $X$ there is another family of subsets $\mathcal B$ where $B \in \mathcal B$ iff it is equal to the union of elements in $A$. Let $f$ be a function which takes any collection of subsets of $X$ and returns the collection of all unions of those subsets as an output.
If you apply $f$ to a basis $\mathcal A$ you will get back a topology of $X$. It is in this sense that we say a basis generates exactly one topology and we call $\mathcal A$ a basis for $f(\mathcal A)$
Conversely, if you already have a topology $\tau$ on the space, then you may be able to find a basis $\mathcal A$ such that $f(\mathcal A) = \tau$. This basis will almost never be unique as a topology is its own basis.
