I am solving Exercise 4.1, Question 17(v) from Topology without Tears (link) by Sidney Morris. (This exercise is marked with a star.)
Let $S = \{ \frac{1}{n} \,:\, n \in \mathbb N \}$. Define a set $C \subseteq \mathbb R$ to be closed if $C = A \cup T$ where $A$ is closed in the euclidean topology on $\mathbb R$ and $T$ is any subset of $S$. [Show that] The complements of these closed sets form a topology $\mathcal T$ on $\mathbb R$ which is Hausdorff but not regular.
We need to show three things here: $\mathcal T\ $ is a topology, it is Hausdorff, but it is not regular. I can prove the "Hausdorff but not regular" part, but I am confused about showing that this is actually a topology!
Here is my attempt. (I will put quotes around the words open and closed, whenever they are with respect to $\mathcal T\ $, to remind myself that I haven't yet verified that $\mathcal T\ $ is a topology.) Clearly I should just verify that $\mathcal T\ $ satisfies the topology axioms.
I can see why both $\mathbb R$ and $\emptyset$ are "closed" sets in $\mathcal T\ $.
I can also show that if $C_1 = A_1 \cup T_1$ and $C_2 = A_2 \cup T_2$ are two "closed" sets, then their union is also "closed". Hence, by induction, a finite union of "closed" sets is also "closed".
The part that I am stuck in is showing that an arbitrary intersection of "closed" sets is also closed. Let $I$ be an arbitrary index set. For $i \in I$, let $C_i = A_i \cup T_i$ be such that $A_i$ is closed in $\mathbb R$ and $T_i \subseteq S$. I want to write the intersection
$ C := \bigcap_{i \in I} (A_i \cup T_i) $ in a form which makes it evident that $C$ is "closed". But naively distributing the $\bigcap$ over the $\cup$ does not seem to work. Please suggest some hints!
Though this is not really homework, I'll add the homework tag since I am not looking for complete solutions anyway.