I try to figure out the equivalence between the definitions of basis for topology given in Munkres 2nd edition and wikipedia.
In wikipedia;
A base is a collection $\mathcal{B}$ of subsets of $X$ satisfying the following properties:
The base elements cover $X$.
Let $B_1$, $B_2$ be base elements and let $I$ be their intersection. Then for each $x$ in $I$, there is a base element $B_3$ containing $x$ and contained in $I$.
Now, If I take $X$ = { a, b, c } and $\mathcal{B}$ = { {a, b, c} } then , is this a basis for any topology?
Since,
i) Element of $\mathcal{B}$ covers $X$.
ii) There are not such $B_1$ and $B_2$ as in definition of basis !
My question is; Does this example satisfy the definition of basis? If yes, Is it necessary to mention the associated topology with basis?
Yes, it is a basis for precisely the reason you stated. Namely, it satisfies the definition of a basis.
Regarding whether or not you need to state whatever topology, that would be up to personal taste in my opinion. However in many cases you describe a topology in terms of a basis for that topology such as saying that the euclidean topology of the plane is the topology generated by the basis of open balls.