I would like to clarify the definition of the co-finite topology. The general definition says this:
Let $X$ be a non empty set. Then the collection of subsets of $X$ whose compliments are finite along with the empty set forms a topology on $X$, and is called the co-finite topology.
There is also the example of this statement:
$$
\tau = \{\varnothing, \{1\}, \{2\}, \{3\}, \{1,2\}, \{2,3\}, \{1,3\}, X\}
$$
which is a co-finite topology because the compliments of all the subsets of X are finite.
Generally, I want to ask one question: why is the complement of each subset of $X$ finite? For example subset which contains only $\{1\}$ for example, complement of this subset is all number except $1$ right? then why is this finite?
Best Answer
Check, that in order, that $\tau$ is the cofinite topology, the complement of $\{1\}$ has to be finite, so $X$ has to be finite.
On a finite set, the cofinite topology is simply $2^{X}$, the discrete topology, since any subset has a finite complement. So, if $\tau$ is a cofinite topology, then $X=\{1,2,3\}$. If $X$ is supposed to be some other set, then $\tau$ is not the cofinite topology.