# The Stacks Project

## Tag 0065

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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