I've been trying to figure out what I can say about the following definition:
https://en.wikipedia.org/wiki/Partially_ordered_space
A topological space $X$ with a partial order $\leq$, is called a partially ordered space if it's graph $Gr(\leq)\subset X\times X$, is a closed subset of $X\times X$.
Are the following sets open/closed in $X$?:
$l(x)=\{y \in X: x\leq y \}$ and $r(x)=\{y \in X: y\leq x \}$
In general, for a topological space $X$ with a binary relation whose graph is closed, can I say that $l_R(x)=\{ y\in X: (x,y)\in R \}$ or $r_R(x)=\{ y\in X: (y,x)\in R \}$ are closed/ open?
Consider following embeddings
$$\alpha(x):X\to X\times X$$ $$\alpha(x)(y)=(x,y)$$ $$\beta(x):X\to X\times X$$ $$\beta(x)(y)=(y,x)$$
Both are continuous for any $x$. Therefore if $Gr(\leq)$ is closed then so is the preimage. But $\alpha(x)^{-1}(Gr(\leq))=l(x)$ and $\beta(x)^{-1}(Gr(\leq))=r(x)$. Proving that both subsets are closed.
The choice of $Gr(\leq)$ is clearly irrelevant in this context. It will work for any binary relation which is closed in $X\times X$. Also it will work if we switch "closed" to "open".