I am trying to verify that the following satisfies the axioms for a neighborhood system:
Let $(X,\leq)$ be a partially ordered set.
Define $S_r(x)=\{y\mid x\leq y\}$ and $\mathcal{U}_x=\{U(x)\mid S_r(x)\subseteq U(x)\}$. Then $\mathcal{U}_x$ is a neighborhood system at $x$. In particular, I am trying to verify the fourth axiom:
(iv) If $U(x)\in\mathcal{U}_x$, then there exists $V\in \mathcal{U}_x$ such that if $y\in V$, then $U(x) \in \mathcal{U}_y$.
So I assume that $U(x)\in \mathcal{U}_x$. This means that $S_r(x)\subseteq U(x)$. I need to choose $V\in \mathcal{U}_x$ satisfying the property. This is where I get stuck.
Try $V = x^\uparrow = \{y: x \le y\}$. By the definition of the neighbourhood system we know that $x^\uparrow \subseteq U(x)$, If $y \in V$ it's clear that $y^\uparrow \subseteq x^\uparrow \subseteq U$ and so $U \in \mathcal{U}_y$