Harald, as Peter May observes in his notes on finite topological spaces (but as must already be standard), your pre-ordered sets provide the only examples. Indeed, if $X$ is an Alexandrov space (or Alexandrov-discrete space, whatever the terminology is), then we may impose a pre-order on it by demanding for $x, y \in X$ that $x \le y$ if and only if $x$ lies in every neighbourhood of $y$. This becomes a genuine partial order exactly when $X$ is $T_0$.

(Sorry; I didn't realise until after posting that Joel had already said this.)

Harald, as Peter May observes in his notes on finite topological spaces (but as must already be standard), your pre-ordered sets provide the only examples. Indeed, if $X$ is an Alexandrov space (or Alexandrov-discrete space, whatever the terminology is), then we may impose a pre-order on it by demanding for $x, y \in X$ that $x \le y$ if and only if $x$ lies in every neighbourhood of $y$. This becomes a genuine partial order exactly when $X$ is $T_0$.

(Sorry; I didn't realise until after posting that Joel had already said this.)

Harald, as Peter May observes in his notes on finite topological spaces (but as must already be standard), your partially-ordered sets provide the only examples. Indeed, if $X$ is an Alexandrov space (or Alexandrov-discrete space, whatever the terminology is), then we may impose a partial order on it by demanding for $x, y \in X$ that $x \le y$ if and only if $x$ lies in every neighbourhood of $y$.

Harald, as Peter May observes in his notes on finite topological spaces (but as must already be standard), your pre-ordered sets provide the only examples. Indeed, if $X$ is an Alexandrov space (or Alexandrov-discrete space, whatever the terminology is), then we may impose a pre-order on it by demanding for $x, y \in X$ that $x \le y$ if and only if $x$ lies in every neighbourhood of $y$. This becomes a genuine partial order exactly when $X$ is $T_0$.

(Sorry; I didn't realise until after posting that Joel had already said this.)