A topological space is defined as a non-empty set $X$ together with a given collection of subsets $T$ (topology) of $X$, such that,
(i) any union of these subsets is one of the subsets.
(ii) any finite intersection of subsets is one of the subsets.
(iii) these collection of subsets contain $\emptyset$ and $X$.
Thus the pair $(X,T)$ together defines a topological space.
Now my question is, if a manifold $M$ is defined as a topological space then what are the collection of subsets of $M$ that specifies the generic topology $T$ in a generic $M$?