Consider $(X,\prec)$ partial ordered set and consider the collection of set $\{U_L(x) \}_{x \in X}$ where, $U_L(x) = \{y\in X: y \prec x \}$ . It is easy to prove this is a base for some topology over $X$, but this topology has(or maybe not) an interesting property that I can't prove.
"Every arbitrary intersection of open sets is a open set."
I'm trying with arbitrary intersections of $U_L(x)$ but I think that not have a bound to rewrite as an $U_L(z)$ for some $z\in X$
This problem is on Dugundji Topology Book (Cap III, sect 3)
Let $\mathscr{V}$ be a non-empty family of open sets, and suppose that $x\in\bigcap\mathscr{V}$. Then for each $V\in\mathscr{V}$ there is an $x_V\in X$ such that $x\in U_L(x_V)\subseteq V$, i.e., such that $x\prec x_V$. If $y\prec x$, then $y\prec x_V$ for each $V\in\mathscr{V}$, so
$$y\in\bigcap_{V\in\mathscr{V}}U_L(x_V)\subseteq\bigcap\mathscr{V}\;,$$
and hence $x\in U_L(x)\subseteq\bigcap\mathscr{V}$. Thus, $U_L(x)$ is an open nbhd of $x$ contained in $\bigcap\mathscr{V}$, and since $x$ was an arbitrary point of $\bigcap\mathscr{V}$, it follows that $\bigcap\mathscr{V}$ is open.