Definition 5.19.4. Let $f : X \to Y$ be a continuous map of topological spaces.

We say that

*specializations lift along $f$*or that $f$ is*specializing*if given $y' \leadsto y$ in $Y$ and any $x'\in X$ with $f(x') = y'$ there exists a specialization $x' \leadsto x$ of $x'$ in $X$ such that $f(x) = y$.We say that

*generalizations lift along $f$*or that $f$ is*generalizing*if given $y' \leadsto y$ in $Y$ and any $x\in X$ with $f(x) = y$ there exists a generalization $x' \leadsto x$ of $x$ in $X$ such that $f(x') = y'$.

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