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

1. 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: $x' \leadsto x$.

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

3. 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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