Semicontinuous ordered topological space which is not continuous ordered

Question

I'm reading article Mobs, trees, and fixed points by J. E. Ward. A partially ordered topological space (POTS) is defined to be a space $X$ together with a partial order $\le$ defined on $X$ such that $\le$ is semicontinuous in the sense that

$$L(x)=\{a\in X:a\le x\}\quad\text{and}\quad M(x)=\{a\in X:x\le a\}$$

are closed sets for each $x\in X$. The partial order is continuous if it has a closed graph in $X\times X$.

What is an example of POTS whose associated partial order is not continuous?

Answer 1

The line with two origins is an example. Denote the space by $X$, let $0$ and $0'$ be the two origins, and let $\preceq$ be the natural partial order on $X$:

• if $x,y\in X\setminus\{0,0'\}$, then $x\preceq y$ iff $x\le y$ in $\Bbb R$;
• if $x\in X\setminus\{0,0'\}$ and $y\in\{0,0'\}$, then $x\preceq y$ iff $x<0$ in $\Bbb R$, and $y\preceq x$ iff $0<x$ in $\Bbb R$; and
• $0\not\preceq 0'\not\preceq 0$.

It’s easy to verify that $X$ is a POTS. To see that $X$ is not a continuous POTS, let $G$ be the graph of $\preceq$, and note that $\langle 0,0'\rangle\notin G$. Let $U$ be any open nbhd of $\langle 0,0'\rangle$ in $X\times X$; then there are open sets $V$ and $W$ in $X$ such that $\langle 0,0'\rangle\in V\times W\subseteq U$. Clearly there is an $\epsilon>0$ such that

$$(-2\epsilon,0)\cup(0,2\epsilon)\subseteq V\cap W\;.$$

But then $\langle-\epsilon,\epsilon\rangle\in U\cap G$, so $\langle 0,0'\rangle\in(\operatorname{cl}G)\setminus G$, and $G$ is not closed in $X\times X$.

You’ll notice that $X$ is not Hausdorff: this is necessary, since it’s not hard to show that a Hausdorff POTS is continuous. And it’s clear that every POTS is $T_1$, so all examples of the kind that you want will be $T_1$ spaces that are not Hausdorff.