Suppose $X$ is a real line with usual topology, and let $PT(X)$ be a category of all elements (points) of $X$ with mapping between points as a morphism. Let TOP be a category of all topological spaces on a countably infinite set and continuous mappings as a morphism.
Let $F$ be a functor sending object of $PT(X)$ to object in TOP, which associates each point of $X$ to a topological space. Is it okay to endow $F$ with additional structure that for any two objects $x$ and $y$ in $PT(X)$, $x + y$ implies $F(x)$ intersects $F(y)$ ? $x + y$ is of course another element of $X$ and we know there exists intersection topology between objects of TOP.
$F(x)=F(x+0)=F(x)\cap F(0)\subseteq F(0)=F(x-x)=F(x)\cap F(-x)\subseteq F(x)$ would imply $F(x)=F(0)$ for all $x$.