Find a subset of $\mathbb{R}^2$ that is path connected but is locally connected at none of its points.

Question

I'm doing this exercise:

Let $X$ denote the rational points of the interval $[0,1]\times\{0\}$ of $\mathbb R^2$. Let $T$ denote the union of all line segments joining the point $p=(0,1)$ to points of $X$.

1. Show that $T$ is path connected, but is locally connected only at the point $p$.
2. Find a subset of $\mathbb R^2$ that is path connected but is locally connected at none of its points.

I've no problems with part a. But I'm getting stuck with part b. Any hints/ideas how I can construct such a subset ?

Answer 1

Let $T$ be a previous space, and define $S$ as follows: Let $Y$ denote the set of rational points of the interval $[0,1]\times \{1\}$. Let $S$ denote the union of all line segments between the point in $Y$ and the point $q=(1,0)$. Consider $S\cup T$.

Answer 2

Part b) can be done using similar ideas as part a). define $T = \{[(q,0),(0,1)]|q\in \mathbb{Q}\}$ $\bigcup$ $\{[(q,1),(1,0)]|q\in \mathbb{Q}\}$ where $[x,y]$ denotes the line joining $x$ and $y$. Then $T$ is path-connected as it is the union of path-connected line segment $p = [(0,1),(1,0)]$ and the union of other line segments all of which intersects $p$. But it is not locally connected at any of its point due to similar reason as in part a).