What is a typical basis for the finite complement topology $\cal{T}_f$? (James R. Munkres "Topology 2nd Edition") I am reading "Topology 2nd Edition" by James R. Munkres.
On p.77, Example 3:

Let $X$ be a set; let $\cal{T}_f$ be the collection of all subsets $U$ of $X$ such that $X-U$ either is finite or is all of $X$. Then $\cal{T}_f$ is a topology on $X$, called the finite complement topology.

I wonder what a typical basis for $\cal{T}_f$ is.
I only know $\cal{T}_f$ is a basis for $\cal{T}_f$.
Please tell me a typical basis for $\cal{T}_f$.
 A: $\cal{T}_f$ is a typical basis for $\cal{T}_f$.
If $X$ is finite then $\big\{\{x\}\big\}_{x\in X}$ will be a basis for $\cal{T}_f$, i.e. $(X,\cal{T}_f)$ is discrete when $X$ is finite.
For any $n\in\mathbb{N}$ the set $\big\{ A\subseteq X\ \big|\ n\leq |X-A|<\infty\big\}$ will be a basis for $\cal{T}_f$. And these are pairwise different if $X$ is infinite.
A: The only natural bases for the cofinite topology $\tau_f$ on an infinite set $X$ are $\tau_f$ and $\tau_f\setminus\{\varnothing\}$, the latter being more common. There are other bases, however. For instance, for any $n\in\Bbb Z^+$ the family
$$\mathscr{B}_n=\{X\setminus F:F\subseteq X\text{ and }n\le|F|<\omega\}$$
of complements of finite sets of cardinality at least $n$ is a base for $\tau_f$. To see this, let $U=X\setminus F\in\tau_f$ with $|F|<n$. Let $D_0$ and $D_1$ be disjoint subsets of $X\setminus F$ such that $|D_0|=|D_1|=n-|F|$, and let $F_i=F\cup D_i$ for $i=0,1$. Then
$$U=X\setminus F=X\setminus(F_0\cap F_1)=(X\setminus F_0)\cup(X\setminus F_1)\,,$$
where $X\setminus F_i\in\mathscr{B}_n$ for $i=0,1$.
