Examples of locally connected sets What are some examples of locally connected sets? I am trying to understand what it means for the Mandelbrot set to be locally connected.
 A: Take for instance:
$\bullet$$\mathbb{R}^n$ is a locally connected space.
$\bullet$ An interval in $\mathbb{R}$ is a locally connected set with respect to the standard topology.
$\bullet$ Or a union of intervals,for example $[0,1] \cup[3,4]$ with respect to the standardo topology.
We know that a path connected set in $\mathbb{R}^n$ is connected,thus a locally path connected in $\mathbb{R}^n$ is locally connected.
Thus $[0,1] \cup [3,4]$ is locally path connected thus locally connected.
$\bullet$ Take the open unit disc in $\mathbb{R}^2$  namely $B(0,1)=\{(x,y) \in \mathbb{R}^2|x^2+y^2<1\}$ or an open rectangle.
$\bullet$ Take a discrete topological space.
A discrete space $X$ is totally disconnected but it is locally connected because for every $x \in X$ the noeighborhood $\{x\}$ is connected.
$\bullet$ Take the set $X=\{x\}$ with the topology $T=\{\{x\},\emptyset\}$.
Hope this helps.
A: Maybe it would also be useful to see an example of a connected set that is not locally connected: the topologist's sine curve $(\{0\} × [-1, 1]) ∪ \{(x, \sin\frac{1}{x}): x ∈ (0, 1]\}$ is such a space. It is connected, but not locally connected at the points $\{0\} × [-1, 1]$. For example the point $(0, 0)$ cannot be separated by clopen sets from points $(\frac{1}{kπ}, 0)$, but this cannot be witnessed locally.
