Let $(P,\leq)$ be a poset with more than $1$ point such that the [interval topology](https://mathoverflow.net/questions/215673/properties-of-the-interval-topology-of-the-lattice-of-functions) $\tau\_i(P)$ is path-connected and $T\_2$. Does this imply that $[0,1]$ order-embeds into $P$?


(This is a follow-up question to [Path-connected interval topologies](https://mathoverflow.net/questions/215693/path-connected-interval-topologies) to which Ramiro de la Vega gave a concise argument that I had overlooked.)



Path-connected Hausdorff interval topologies
There is a counterexample as a subset of the real plane endowed with the natural partial order.


Take the graph $\Gamma$ of the standard continuous monotone map $f:C\to[0,1]$ of the Cantor set $C\subset[0,1]$ onto $[0,1]$. The map $f$ is defined anlytically by $f:\sum\_{n=1}^\infty \frac{2x\_n}{3^n}\mapsto \sum\_{n=1}^\infty \frac{x\_n}{2^n}$ where $x\_n\in\{0,1\}$.


Let $Q\_2$ be the set of binary-rational number in the open interval $(0,1)$. By the construction of $f$, for any $y\in Q\_2$ the preimage $f^{-1}(y)$ consists of two points and for $y\in[0,1]\setminus Q\_2$ the preimage is a singleton. It follows that $\Gamma\_0=\Gamma\cap (\mathbb R\times Q\_2)$ is a countable dense subset of $\Gamma$.


Now let $\rho$ be the clockwise rotation of the real plane on $\pi/4$. Consider the line $L=\{(x,y):x+y=0\}$ and the subset $X=\rho(\Gamma)\cup(L+\rho(\Gamma\_0))$. It can be shown that $X$ is a required path-connected ordered pospace containing no ordered arc. (To prove this property of $X$ use the argument of Baire category).


