It is claimed in the Encyclopedia of Mathematics that $S^1$ is an ANR. However this seems to be wrong as there is no retraction from the disc onto the circle. What am I missing?
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[I'll assume you're talking about the metric space variety of ARs and ANRs]
You seem to be confused between the definition of a space being an AR (absolute retract) and an ANR (absolute neighbourhood retract). In fact, a metric space is an AR if and only if it is contractible and an ANR.
An ANR is a metric space $X$ such that for any other metric space $Y$ for which $X$ is a closed subset of $Y$, there exists an open subset $X \subset U\subset Y$ such that $X$ is a retract of $U$.
In the case of a circle $X=S^1$ and $Y=D^2$ the disk, we can take $U = D^2 \setminus \{(0,0)\}$ and then $S^1$ is a retract of $U$ under the usual radial projection map.
It's well known that every topological manifold is an ANR.
Dan Rust
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