I am trying this problem.
4-10(From (J.Lee) Intoduction to Topological manifold)
Let $S$ be the square $I\times I$ with the order topology generated by the dictionary order (see Problem 4-6).
(a) Show that $S$ has the least upper bound property.
(b) Show that $S$ is connected.
(c) Show that $S$ is locally connected, but not locally path-connected.
I have done for (a) and (b) and am stuck at $(c)$.
My trial is ....
Recall that the order topology generated by the dictionary order $\leq $ is the topology that generated by sets \begin{align*} &G_{(a,b)}=\{(x,y)\in S : (x,y) > (a,b) \}&L_{(a,b)}=\{(x,y) \in S: (x,y) < (a,b) \} \end{align*}
for each $(a,b)\in I\times I$.
(c)
Let $p\in S$ be given and let a neighborhood $U\subseteq S$ of $p$ be given. Note that $L_p\cap U_p=\{p\}\subseteq U$ and it is open since each of them are two of the generator of this topology and the union is a finite intersection of open sets. Note that singleton set is clearly connected since we cannot find any two disjoint nonempty set of it. Thus, $S$ is locally connected.\
However, if it is true, then it is also locally connected since singleton set is also path connected... which is inconsistent with the problem that I need to prove. I don't see where I am wrong.
And help would be appreciated.