= Continuous maps from an infinite cofinite space to a Hausdorff space are constant
If $X$ is an <infinite set> with its <cofinite topology> and $Y$ is a <Hausdorff space>, every <continuous map> $f:X\to Y$ is a <constant function>. Otherwise two distinct image points would have <disjoint> <open> <neighbourhoods>. Their <inverse images> would be nonempty <disjoint> <open sets> of $X$, contradicting its <hyperconnected space> property.
For example, $f(n)=n$ from $\mathbb N_{>0}$ with its <cofinite topology> into the <real line> with its usual <Euclidean metric> is not <continuous>: the <inverse image> of the <open interval> $(1/2,3/2)$ is $\{1\}$, which is not <open> in the infinite domain. A <continuous> real-valued readout on that domain can only report one value. Finite cofinite domains are different: their <discrete topology> allows arbitrary <functions> into any <topological space>.
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