Definition 5.19.1. Let $X$ be a topological space.

If $x, x' \in X$ then we say $x$ is a

*specialization*of $x'$, or $x'$ is a*generalization*of $x$ if $x \in \overline{\{ x'\} }$. Notation: $x' \leadsto x$.A subset $T \subset X$ is

*stable under specialization*if for all $x' \in T$ and every specialization $x' \leadsto x$ we have $x \in T$.A subset $T \subset X$ is

*stable under generalization*if for all $x \in T$ and every generalization $x' \leadsto x$ we have $x' \in T$.

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