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Consider dictionary order topology in $\mathbb{R}^2$.

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

Show that $T$ is path connected.

Well, that is easy: Note that every line segments in $T$ are connected with $p$. Hence path connected.$\tag{**}$


But: Consider $X=[0,1]\times 0$ in $\mathbb{R}^2$ and $T$ denotes the union of all line segments joining the point $p=0\times 1$ to points of $X$.

In this case, I can use the same argument as before and claim that $T$ is path connected. But clearly $T$ is not path connected for the fact $[0,1]^2$ is not path connected.

So, is my argument (**) is vague? Please someone help!

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  • $\begingroup$ Why do you think the standard square is not path connected? $\endgroup$ – M. Van Sep 20 '17 at 14:10
  • $\begingroup$ @M.Van It is shown in Munkres Topology that ordered square is connected but not path connected (Example 6, page 156). That is why I did not wrote that. $\endgroup$ – topology_001 Sep 20 '17 at 14:14
  • $\begingroup$ Ah I see, not the standard topology on the square. $\endgroup$ – M. Van Sep 20 '17 at 14:28
  • $\begingroup$ Could you give the topology on each of the topological spaces you talk about? $\endgroup$ – M. Van Sep 20 '17 at 14:30
  • $\begingroup$ @M.Van it is dictionary order topology. Otherwise it would be mentioned in the question.(it is mentioned that if nothing written then dictionary order topology.) $\endgroup$ – topology_001 Sep 20 '17 at 14:34
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As I understood, $T$ consists of the segments $[x, p]$, where $x\in [0,1]\times 0$. When $x\ne 0\times 0$, the open segment $(x, p)$ is a discrete subspace in the dictionary order topology, so this don’t imply path-connectedness of $T$. Also, something is wrong, because if you consider on $T$ the order topology induced from $\Bbb R^2$ then $T$ is clearly not path-connected, because if $x>0$ then $x\times [0,1]$ is a countable neighborhood of $x$ in $T$ and we can easily show that any continuous map $f$ from a unit segment $I=[0,1]$ (endowed with the standard topology) to $T$ such that $f(0)=x\times 0$ is constant. So maybe you have to consider on $T$ the order topology induced from $[0,1]^2$, maybe, even generated by a flipped order $a\times b< c\times d$ iff $b<d$ or $b=d$ and $a<c$.

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  • $\begingroup$ From the part "Also, something is wrong,..." --- this is exactly what I am trying to say. I knew that $T$ in the second case is not path connected. $\endgroup$ – topology_001 Sep 22 '17 at 6:33
  • $\begingroup$ Actually I want to know why the argument for the first problem does not work in the next case : (i.e. why the argument when $X$ is countable does not work in the the case of when $X=[0,1]\times 0$. $\endgroup$ – topology_001 Sep 22 '17 at 6:34
  • $\begingroup$ It is just: I am saying "consider any two point in $T$, since they are path connected with $p$, T is path connected". Why this argument do not work in the next case! $\endgroup$ – topology_001 Sep 22 '17 at 6:38
  • $\begingroup$ .... So the conclusion is: If that argument do not work for $X=[0,1]\times 0$, how can I be sure that that argument will work for $X=$ rational points on $[0,1]\times 0$ .... I hope you understand!!! $\endgroup$ – topology_001 Sep 22 '17 at 6:41

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