A $T_1$ space has the property that every open set can be countably exhausted by closed sets iff it is perfectly normal.
Assume that $X$ is $T_1$ and that every $U\in\tau$ can be countably exhausted by closed sets. Let $x\in U\in\tau$. There are sequences $\langle F_n:n\in\Bbb N\rangle$ of closed sets and $\langle V_n:n\in\Bbb N\rangle$ of open sets such that
$$F_n\subseteq V_n\subseteq\operatorname{cl}V_n\subseteq F_{n+1}\tag{1}$$
for each $n\in\Bbb N$ and $U=\bigcup_{n\in\Bbb N}F_n$. Lemma 1.5.14 of Engelking’s General Topology ensures that $X$ is $T_4$ (normal and $T_1$); the proof is similar to the proof that regular Lindelöf spaces are normal.
It’s also clear from $(1)$ that every open set in $X$ is a countable union of regular closed sets. In particular, every open set is an $F_\sigma$, so $X$ is even perfectly normal. It’s well-known that in a $T_1$ space perfect normality is equivalent to each of the following properties:
- Open sets are cozero-sets.
- Closed sets are zero-sets.
- If $H$ and $K$ are disjoint closed sets, there is a continuous $f:X\to[0,1]$ such that $H=f^{-1}[\{0\}]$ and $K=f^{-1}[\{1\}]$.
Conversely, suppose that $X$ is a perfectly normal $T_1$ space, and let $U$ be a non-empty open subset of $X$. There is a continuous $f:X\to[0,1]$ such that $U=f^{-1}\big[(0,1]\big]$. For $n\in\Bbb N$ let
$$F_n=f^{-1}\left[\left[\frac1{2^n},1\right]\right]\text{ and }V_n=f^{-1}\left[\left(\frac1{2^{n+1}},1\right]\right]\;;$$
clearly the sequences $\langle F_n:n\in\Bbb N\rangle$ and $\langle V_n:n\in\Bbb N\rangle$ satisfy $(1)$, and $U=\bigcup_{n\in\Bbb N}F_n$.
The "finite closed topology" describes the closed sets. Which gives you an exact definition of the open sets, they are the complements of closed sets. So indeed this is the co-finite topology (or finite complement topology).
A subset $U$ is open if and only if $U=\varnothing$ or $X\setminus U$ is finite.
Imagining the entire topology is a bit tricky, since infinite sets can be very large, and therefore have many finite sets. So it's best to think about this in terms of definitions. There is a definition when a set is open, and when it is closed. Now check if these definitions meet the requirement of being $T_1$.
And for that matter, allow me to remind you that $X$ is a $T_1$ space if and only if every singleton is closed.
Best Answer
There is no such standard notation. The safest approach is to let $\langle X,\tau\rangle$ (or $\langle X,\mathscr{T}\rangle$, etc.) be a topological space and then explicitly to name the collection of closed sets, e.g., by letting $\mathscr{F}=\{X\setminus U:U\in\tau\}$.
Since $F$ (from French fermé) is one of the letters that I conventionally use for closed sets, I’m likely to use $\mathscr{F}$ or $\mathscr{C}$ (for closed) for the collection of closed sets unless those letters have been pre-empted.