The finite-closed or co-finite Topology

Still on general topology, today i will be writing on a very special type of topology called the co-finite or finite closed topology, in some cases it is also called the third type topology.
Now if you have been following my lectures on general topology you will see that my last lecture was on the "Discrete and Indiscrete Topology ".
Now what do we mean by the cofinite topology?
This is defined as :
Let $X$ be any non-empty set. A topology $\mathcal{T}$ on $X$ is called the finite-closed topology if the closed subsets of $X$ are $X$ and all finite subsets of $X$; i.e the open sets are $\phi$ and all subsets of $X$ which have finite complements.
OR in order words, we can
define the co-finite topology as a set $X$ which is s nonempty set and let $\mathcal{T}$ be a collection of null sets and all subsets of $X$ whose complements are finite, then $\mathcal{T}$ is called cofinite topology on $X$, and the pair $(X,\mathcal{T})$ is called co-finite topological space.

Examples of co-finite Topology
Let $X=\{a,b,c,d\}$, find the co-finite topology on $X$.
Solution

Since the set $X$ is given as $X=\{a,b,c,d\}$ then we generate a power set for $X$, now to generate a power set for $X$,we use the formula $P(X)=2^n$, where $n$ is the cardinality of $X$.
i.e $P(X)=2^n=2^4=16$ because $n=4$.
so $P(X)=\{\phi,X,\{a\},\{b\},\{c\},\{d\},\{a,b\},\{a,c\},\{a,d\},\{b,c\}$,
$ \{c,d\},\{b,d\}, \{a,b,c\},\{a,b,d\},\{a,c,d\},\{b,c,d\}\}$
Now we take the compliments of each of the elements in $X$.
$\phi^c=X-\phi=X\in{\mathcal{T}}$
$X^c=X-X=\phi\in{\mathcal{T}}$
$\{a\}^c=\{b,c,d\}\in{\mathcal{T}}$
$\{b\}^c=\{a,c,d\}\in{\mathcal{T}}$
$\{c\}^c=\{a,b,d\}\in{\mathcal{T}}$
$\{d\}^c=\{a,b,c\}\in{\mathcal{T}}$
Now since the compliments of the elements in $X$ are in $\mathcal{T}$, then we say $\mathcal{T}$ is a topology on $X$.