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I have been looking for an example of a metrizable space that is not an absolute neighborhood retract (ANR).

Recall that a metrizable space $X$ is called an ANR if there exists an open set $U$ in a metrizable space $Y$ and an embedding $f: X \to Y$ such that $f(X)$ is a retract of $U$.

Equivalently, $X$ is called an ANR if for each closed subset $A$ of a metrizable space $Z$ and every continuous map $f: A \to X$, there exists a neighborhood $U$ of $A$ in $Z$ and a continuous map $g: U \to X$ that extends $f$, i.e., $g_{|A} = f$.

I understand that all metrizable topological manifolds are metric ANRs. However, I'm unable to find an example that is not an ANR. Can someone please suggest a (possibly exotic but preferably well-known) example of a metric space that is NOT a metric ANR?

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    $\begingroup$ All ANRs are locally contractible. There are plenty of subsets of the plane (or line) which are not. For example, the Cantor set, sine Curve, Warsaw circle, etc... $\endgroup$
    – Tyrone
    Commented Jan 19 at 13:37
  • $\begingroup$ Your definition of ANR is wrong. $\endgroup$
    – Moishe Kohan
    Commented Jan 19 at 15:38
  • $\begingroup$ @MoisheKohan I am simply following Definition 5.3 and Remark 5.3 from sciencedirect.com/topics/mathematics/…. $\endgroup$
    – Katrina
    Commented Jan 19 at 16:51
  • $\begingroup$ No, you are not following the standard definition: A "metrizable space" is not the same thing as a "normed space." $\endgroup$
    – Moishe Kohan
    Commented Jan 19 at 16:53

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