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