The Stacks Project


Tag 0063

Chapter 5: Topology > Section 5.19: Specialization

Definition 5.19.3. Let $f : X \to Y$ be a continuous map of topological spaces.

  1. We say that specializations lift along $f$ or that $f$ is specializing if given $y' \leadsto y$ in $Y$ and any $x'\in X$ with $f(x') = y'$ there exists a specialization $x' \leadsto x$ of $x'$ in $X$ such that $f(x) = y$.
  2. We say that generalizations lift along $f$ or that $f$ is generalizing if given $y' \leadsto y$ in $Y$ and any $x\in X$ with $f(x) = y$ there exists a generalization $x' \leadsto x$ of $x$ in $X$ such that $f(x') = y'$.

    The code snippet corresponding to this tag is a part of the file topology.tex and is located in lines 3246–3259 (see updates for more information).

    \begin{definition}
    \label{definition-lift-specializations}
    Let $f : X \to Y$ be a continuous map of topological spaces.
    \begin{enumerate}
    \item We say that {\it specializations lift along $f$} or that $f$ is
    {\it specializing} if given $y' \leadsto y$ in $Y$ and any $x'\in X$ with
    $f(x') = y'$ there exists a specialization $x' \leadsto x$ of $x'$ in $X$ such
    that $f(x) = y$.
    \item We say that {\it generalizations lift along $f$} or that $f$ is
    {\it generalizing} if given $y' \leadsto y$ in $Y$ and any $x\in X$ with
    $f(x) = y$ there exists a generalization $x' \leadsto x$ of $x$ in $X$ such
    that $f(x') = y'$.
    \end{enumerate}
    \end{definition}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 0063

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?