There are abundant counterexamples in literature of the $2$ statements –
- $X$ is Path Connected $\implies$ $X$ is Locally Path Connected
- $X$ 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 $2$?
Best Answer
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 $X$.
Informally, we attach a ‘sticker’ in the form of a copy of the closed unit interval to each point on the bottom and top edges of the lexicographically ordered square, and we identify the free ends of the stickers.
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 $\langle x,1\rangle\sim 0^x$.