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

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

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

Proof. Let $y' \leadsto y$ be a specialization in $Y$ where $y'\in f(T)$ and let $x'\in T$ such that $f(x') = y'$. Because specialization lift along $f$, there exists a specialization $x'\leadsto x$ of $x'$ in $X$ such that $f(x) = y$. But $T$ is stable under specialization so $x\in T$ and then $y \in f(T)$. Therefore $f(T)$ is stable under specialization.

The proof of (2) is identical, using that generalizations lift along $f$. $\square$

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