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Tag 0065

Chapter 5: Topology > Section 5.19: Specialization

Lemma 5.19.5. 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. Omitted. $\square$

    The code snippet corresponding to this tag is a part of the file topology.tex and is located in lines 3273–3284 (see updates for more information).

    \begin{lemma}
    \label{lemma-lift-specializations-images}
    Let $f : X \to Y$ be a continuous map of topological spaces.
    \begin{enumerate}
    \item If specializations lift along $f$, and if $T \subset X$
    is stable under specialization, then $f(T) \subset Y$ is
    stable under specialization.
    \item If generalizations lift along $f$, and if $T \subset X$
    is stable under generalization, then $f(T) \subset Y$ is
    stable under generalization.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Omitted.
    \end{proof}

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