= Continuous real maps from the initial-segment topology
Give $\mathbb N$ the <topology> whose nonempty <open sets> are $\mathbb N$ and the finite initial segments. Every nonempty <open set> contains $1$, so any two nonempty <open sets> intersect. A <continuous map> from this <topological space> into a <Hausdorff space> is constant: distinct image points would have disjoint <open> <neighbourhoods>, whose inverse images would be disjoint nonempty <open sets>. This includes <continuous> real-valued maps.
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