Topology

en · official · sha256:1d6ee8a1b98bc09677428df262ad2123d411e5bb168e3ac20a1c67cb0dd9d0a5

\begin{definition} \label{definition-specialization} Let $X$ be a topological space. \begin{enumerate} \item If $x, x' \in X$ then we say $x$ is a {\it specialization} of $x'$, or $x'$ is a {\it generalization} of $x$ if $x \in \overline{\{x'\}}$. Notation: $x' \leadsto x$. \item A subset $T \subset X$ is {\it stable under specialization} if for all $x' \in T$ and every specialization $x' \leadsto x$ we have $x \in T$. \item A subset $T \subset X$ is {\it stable under generalization} if for all $x \in T$ and every generalization $x' \leadsto x$ we have $x' \in T$. \end{enumerate} \end{definition}

Referencing norms: LocallyConstantAlongSpecialization, OpenByGeneralizationStability, generalizes, point_of

en · official · sha256:881627a4bc7da511bfc2e3d7cd01f971abb8e2f72ae02f0a295a59b98de782f8

\begin{lemma} \label{lemma-open-closed-specialization} Let $X$ be a topological space. \begin{enumerate} \item Any closed subset of $X$ is stable under specialization. \item Any open subset of $X$ is stable under generalization. \item A subset $T \subset X$ is stable under specialization if and only if the complement $T^c$ is stable under generalization. \end{enumerate} \end{lemma}

Referencing norms: CoversByPoints, IntersectionByPoints, OpenByGeneralizationStability, SubsetByPoints, candidate_subset, contains, finite_space, target