Lemma 5.19.6. Let $f : X \to Y$ be a continuous map of topological spaces.

If specializations lift along $f$, and if $T \subset X$ is stable under specialization, then $f(T) \subset Y$ is stable under specialization.

If generalizations lift along $f$, and if $T \subset X$ is stable under generalization, then $f(T) \subset Y$ is stable under generalization.

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