 Quotient topology of a cofinite space

A quotient space of a space carrying the cofinite topology has a topology contained in the cofinite topology on its underlying quotient set. If $q:X\to Y$ is the quotient map and $U$ is a nonempty open subset of $Y$, its preimage has finite complement. Since $q$ is surjective, the complement of $U$ is the image of that finite complement, hence is finite. Equality need not hold: partition $\mathbb{Z}$ into its even and odd integers. Both classes are infinite, so their two-point quotient space has the indiscrete topology, whereas the cofinite topology on a finite set is the discrete topology.