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

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 ?

• For 2., have a look at en.wikipedia.org/wiki/Topologist%27s_sine_curve ! This might give you a satisfying example. Mar 18 '14 at 12:48
• @jibounet Topologist's sine curve is locally connected at some points. Mar 18 '14 at 12:49
• You're right, sorry ! Do not take my comment into account. Mar 18 '14 at 12:50
• I suspect the idea is to make use of $T$. What happens to local connectedness at $p$ if we stack a unit vertical translation of $T$ on top of $T$? "It's Turtles all the way up." Mar 18 '14 at 12:56
• I really like William Elliot's answer to this question here. Mar 20 '18 at 13:52

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 segment between the point in $Y$ and the point $q=(1,0)$. Consider $S\cup T$.
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).