I was going through the definition given in J.R munkres:
According to the definition, it seems to me that for every set $X$, there is a unique Co-Finite. Because it has been specified that collection of 'ALL' subsets of $X$ such that…
I Had another question: If we take $X$ to be $= \{a,b,c\}$, then will a topology$=\{\{a\},X,\{\}\}$ be a Co-finite topology on $X$? because if it is, then symmetrically $\{\{b\},X,\{\}\}$ should also be cofinite, which defies the uniqueness.
Yes, it is unique, for the reason that you mentioned.
And $\bigl\{X,\{a\},\emptyset\bigr\}$ is not the co-finite topology on $\{a,b,c\}$ because, for instance, $\{a,b\}\bigl(=X\setminus\{c\}\bigr)$ doesn't belong to it, in spite of the fact that $\{c\}$ is finite.