= 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>.
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