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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 ?

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  • $\begingroup$ For 2., have a look at en.wikipedia.org/wiki/Topologist%27s_sine_curve ! This might give you a satisfying example. $\endgroup$ – jibounet Mar 18 '14 at 12:48
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    $\begingroup$ @jibounet Topologist's sine curve is locally connected at some points. $\endgroup$ – Hanul Jeon Mar 18 '14 at 12:49
  • $\begingroup$ You're right, sorry ! Do not take my comment into account. $\endgroup$ – jibounet Mar 18 '14 at 12:50
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    $\begingroup$ 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." $\endgroup$ – hardmath Mar 18 '14 at 12:56
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    $\begingroup$ I really like William Elliot's answer to this question here. $\endgroup$ – Selrach Dunbar Mar 20 '18 at 13:52
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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 $S\cup T$.

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    $\begingroup$ Is my edit correct ? $\endgroup$ – Kasper Mar 18 '14 at 15:18
  • $\begingroup$ @Kasper Yes, you are correct. Thanks! $\endgroup$ – Hanul Jeon Mar 19 '14 at 3:08
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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).

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