# Some Counterexamples on Connectedness

There are abundant counterexamples in literature of the $$2$$ statements -

1. $$X$$ is Path Connected $$\implies$$ $$X$$ is Locally Path Connected
2. $$X$$ is Arc Connected $$\implies$$ $$X$$ is Locally Arc Connected

In all of the counterexamples I've found, they hold as the space is Path/Arc Connected, but is not Locally Connected (for example, the Extended Topologist's Sine Curve and the Closed Infinite Broom).

So, I wish to ask - are there counterexamples of the above $$2$$ statements, if we also assume that $$X$$ is Locally Connected? How about Locally Path Connected for Statement $$2$$?

• Note that a space that is both locally path-connected and connected space is path-connected. So we don't have an example for the "mixed" case. Aug 5, 2020 at 4:42

Let $$Y=[0,1]\times[0,1]$$ with the lexicographic order topology. For each $$x\in[0,1]$$ let $$I_x$$ and $$I^x$$ be copies of $$[0,1]$$ with its usual topology, and for each $$t\in[0,1]$$ let $$t_x$$ and $$t^x$$ be the copies of $$t$$ in $$I_x$$ and $$I^x$$, respectively. For $$x\in[0,1]$$ identify $$\langle x,0\rangle\in Y$$ with $$0_x\in I_x$$ and $$\langle x,1\rangle\in Y$$ with $$0^x\in I^x$$. Then identify all of the points $$1_x$$ and $$1^x$$ to a single point $$p$$ to get the space $$X$$.
Then $$X$$ is path connected and locally connected, but it is not locally path connected at any of the points $$\langle x,0\rangle\sim 0^x$$ or $$\langle x,1\rangle\sim 0^x$$.
• What exactly do you mean by $\langle x,0\rangle\sim 1^x$? Aug 6, 2020 at 6:02
• @IshanDeo: Argh! It was supposed to be $\langle x,0\rangle\sim 0^x$, the point of $X$ that results from the identification of $\langle x,0\rangle\in Y$ and $0^x\in I^x$. I don’t know whether it was a typo or a mental hiccup, but I’m fixing it as soon as I post this comment! Aug 6, 2020 at 7:14
• So, correct me if I'm wrong, but - you added the lines $I_x$ and $I^x$ for the sole purpose of creating a path between any $2$ points of $X$, which did not originally exist in the ordered square. Also, $X$ is not locally path connected for the same reason the ordered square is not. Aug 6, 2020 at 8:22