Topological intuition to continuous preference relation

For a Microeconomics Course, we are going through MWG, and in the lecture we discussed the notion of a continuous preference relation.
A preference relation $succsim$ on a set $X$ is called continuous if
$forall x,yin X$, for all sequences ${x_k}to x,{y_k}to y$ for which we $x_k succsim y_k,forall k$, we have that $x succsim y$. I am trying to understand the choice of words for continuity of the preference relation here: by some steps, using a theorem from Debreu, we can get from a rational and continuous preference relation to a continuous utility function that represents it. However, can we phrase the above definition in such a way that it corresponds to the usual topological notion of continuity? (A function is continuous if every pre-image of an open set is open.)

What spaces and what function should we consider, to find an equivalent topological definition of continuity of a preference relation? I got as far that it must be either some map $f: Xtimes X to X$, or a map $f: Xtimes Xto {0,1}$, with X in the topology induced by $succsim$ and ${0,1}$ in the discrete topology. But where to from here?