A compact set $X$ is only locally conneted in on point but not locally connected in other points in $X$

Question

When study the theorem of boundary correspondence in complex analysis, I come into connact with the concept of Locally Connectedness and several examples.

Definition: A compact set $X\subset\mathbb{C}$ is said to be locally connected at $x$ if for every $\epsilon>0$ , there is a connected open neighbourhood $E$ of $x$ such that $\mathrm{diam}(E)<\epsilon$(diameter). The topology of $X$ is induced by the usual topology of $\mathbb{C}$.

Then I have two questions as follows:

Is there a connceted compact set $X\subset\mathbb{C}$ which contains more than one point satiesfying

1. $X$ is only locally conneted in on point but not locally connected in other points in $X$;
2. $X$ is only not locally conneted in on point butlocally connected in other points in $X$.

Appreciate any help! And if you shall use some theorems, please quote them clearly.

Answer 1

The cone over the Cantor set is locally connected at its vertex only, see Example 129 in Counterexamples in Topology by Steen and Seebach. Exercise 5.22 in Nadler's Continuum Theory asks you to show that the set of points at which a continuum is not locally connected is dense in itself, so your second example does not exist.