= Solution
Any two nonempty <open sets> in the <cofinite topology> on $\mathbb N$ intersect: their complements have finite union, which cannot exhaust $\mathbb N$. Thus this is a <hyperconnected space>. Suppose a <continuous map> $f$ took two distinct values $a$ and $b$. The usual real line is a <Hausdorff space>, so there are disjoint open intervals $U$ and $V$ containing $a$ and $b$. Their inverse images under the <continuous map> are disjoint, nonempty <open sets>, contradicting hyperconnectedness. \b[Every such $f$ is constant.]
Back to article page
