Confusion about notation in this proof I am looking at the following problem from Munkres' topology and a proof previously asked on this forum.

If $\{\tau_\alpha\}$ is a family of topologies on $X$, show that $\cap \tau_\alpha$ is a topology on $X$.

This was the proof given by a user:
For all $\alpha$, $\varnothing \in \tau_\alpha$ and $X \in \tau_\alpha$. So, $\varnothing \in \cap \{\tau_\alpha\}$ and $X \in \cap \{\tau_\alpha\}$. Now, let $\{U_\beta\}_{\beta \in J}$ be an indexed collection of subsets of $\cap \{\tau_\alpha\}$. Then, for all $\alpha$, $\{U_\beta\}_{\beta \in J} \subseteq \tau_\alpha$. This implies that for all $\alpha$, $\cup\{U_\beta\}_{\beta \in J}\in\tau_\alpha$, since $\tau_\alpha$ is a topology. But then, $\cup\{U_\beta\}_{\beta \in J}\in\cap \{\tau_\alpha\}$.
Now, let $U_1, U_2, \ldots U_n$ be subsets of $\cap \{\tau_\alpha\}$. Then, for all $\alpha$, $U_i \in \tau_\alpha$. This implies that for all $\alpha$, $\cap_{i=1}^n U_i \in \tau_\alpha$, since each $\tau_\alpha$ is a topology. But then, $\cap_{i=1}^n U_i \in \cap \{\tau_\alpha\}$. Therefore, $\cap \{\tau_\alpha\}$ is a topology on $X$.
People told this user the proof was correct, but I am confused on the notation. Wouldn't it be more correct to make the following changes in the proof, or am I confused?
For all $\alpha$, $\varnothing \in \tau_\alpha$ and $X \in \tau_\alpha$. So, $\varnothing \in \cap \tau_\alpha$ and $X \in \cap \tau_\alpha$. Now, let $\{U_\beta\}_{\beta \in J}$ be an indexed collection of elements of $\cap \tau_\alpha$. Then, for all $\alpha$, $\{U_\beta\}_{\beta \in J} \subseteq \tau_\alpha$. This implies that for all $\alpha$, $\cup U_{\beta}$ in$\tau_\alpha$, since $\tau_\alpha$ is a topology. But then, $\cup U_{\beta}\in\cap \tau_{\alpha}$.
Now, let $U_1, U_2, \ldots U_n$ be elements of $\cap \tau_{\alpha}$. Then, for all $\alpha$, $U_i \in \tau_\alpha$. This implies that for all $\alpha$, $\cap_{i=1}^n U_i \in \tau_\alpha$, since each $\tau_\alpha$ is a topology. But then, $\cap_{i=1}^n U_i \in \cap \tau_{\alpha}$. Therefore, $\cap {\tau}_{\alpha}$ is a topology on $X$.
I am mostly confused as to why people giving this user feedback said it was completely correct, I feel like the placement of parenthesis in the unedited proof, make the user's proof very incorrect. Am I right? What is going on here?
 A: It seems your point is that the intersection and union of a family of sets should be written as $\cup U_\alpha$ instead of $\cup \{U_\alpha\}$.
The next paragraph is from the ZFC axioms page on wikipedia:
The axiom of union states that for any set of sets $\mathcal{F}$ there is a set $A$ containing every element that is a member of some member of $\mathcal{F}$:
$\forall \mathcal{F} \,\exists A \, \forall Y\, \forall x [(x \in Y \land Y \in \mathcal{F}) \Rightarrow x \in A]$.
That is, you can think of the union as a 'formula' that assigns to a family of sets the union of all the sets in that family. Seen this way, the notation $\cup \{\tau_\alpha;\alpha\in A\}$ makes sense. As in the question they just called the family $\{\tau_\alpha\}$, they also called it this way in the answer, although it would be better to write the index set.
The notation you use is also accepted and more widely used.
A: If $\mathcal S$ is collection of sets, $\bigcap \mathcal S$ is the same as $\bigcap_{A \in \mathcal S} A$. So, in that problem are abusing the notation by writting $\{\tau_\alpha\}$ instead of $\{\tau_\alpha\}_{\alpha \in J}$ for some index set $J$. Thus, in that case $\bigcap \{\tau_\alpha\}$ refers to $\bigcap \{\tau_\alpha\}_{\alpha \in J}$, which is the same as $\bigcap_{\alpha \in J} \tau_\alpha$, as you say.
