Set of open sets notation

A topology $mathcal T$ on a set $X$ is a set of subsets of $X$ satisfying the well-known axioms for "open sets". Thus, letting $mathfrak P(X)$ denote the powerset of $X$, we have $mathcal T subset mathfrak P(X)$.

In my opininon there is no benefit to regard a topology as an indexed collection ${U_i mid iin I}$. Writing it in this form means that we are given a set $I$ of indices and for each $i in I$ an element $U_i in mathfrak P(X)$. The index set $I$ can be any set. You should be aware that such an indexed collection of sets $U_i$ formally is a function $phi : I to mathfrak P(X)$ where we write $U_i = phi(i)$. Note that this allows $U_i = U_{i'}$ for $i ne i'$, i.e. your indexed family of sets $U_i$ may have multiple occurences of the same set.

If we simply say that $mathcal T subset mathfrak P(X)$, then we can regard $mathcal T $ as indexed collection by taking $I = mathcal T$ and letting $phi : mathcal T to mathfrak P(X)$ be the inclusion, i.e. $phi(U) = U$. This may be called the "self-indexing" of $mathcal T$.

Anyway, both approaches are correct, thus it is a matter of taste which you prefer.