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?

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.
