Let $\preceq$ be a linear/total order relation on some set $X$. Clearly, in general we can't find a countable subset $Y \subset X$ such that for all $x \in X$ there exists a $y \in Y$ with $x \preceq y$.
However, if $X$ is a separable topological space and $\preceq$ interacts in a sufficently nice way with the topology, I would expect that we can find a countable subset $Y$ with the desired property above.
Unfortunately, I have no idea where to look or what to expect. Are there keywords or references I can look at?
Background: I want to apply Zorn's Lemma but I only want to check countable linear chains.