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In the early 1970s Pelczynski noticed that the only surjective isometries on $C(K)$ for the following compact Hausdorff space $K$ are $\pm Id$. I believe this was the first such example.

Quoting from Davis in "Separable Banach spaces with only trivial isometries": Let $K$ be a dendrite containing for each $n>2$ exactly one cut-point of degree $n$ so that the cut-points are dense in $K$. Then the only surjective homomorphism of $K$ is the identity." Banach-Stone then yields the result.

My question is: How does one construct $K$?

One of my issues is that I don't know some of the basic definitions. What is the degree of a cut point? Googling has not been very helpful.

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    $\begingroup$ The degree of a cut-point is probably the number of components it separates the space into upon deletion. $\endgroup$
    – mme
    Commented Sep 6, 2021 at 16:58
  • $\begingroup$ @mme. Thanks, that was my guess. $\endgroup$
    – Kevin Beanland
    Commented Sep 6, 2021 at 17:05
  • $\begingroup$ Start with a closed interval of length $1$. Make the center $x$ of the interval a cut point of degree 3 by attaching an interval of length $1/3$ with end point at $x$. In each component of the three intervals you have when you remove $x$, make the center $y$ a cut point of degree $4$ by attaching two intervals of length $1/4$ with end points at $y$. Et cetera. $\endgroup$
    – Bill Johnson
    Commented Sep 6, 2021 at 18:06
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    $\begingroup$ See also the second answer here mathoverflow.net/questions/188707/… $\endgroup$
    – Alessandro Codenotti
    Commented Sep 6, 2021 at 20:24
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    $\begingroup$ What Alessandro said. $\endgroup$
    – Bill Johnson
    Commented Sep 7, 2021 at 3:14

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