I have a problem with the following definition of a order-dense set.
Definition: Suppose $\succ$ is a binary relation on an uncountable set $X$. A subset $Z$ of $X$ is called $\succ$-order dense if for all $x,y \in X$ such that $x \succ y$ exists some $z \in Z$ with $x\succsim z \succsim y$.
So here there is my problem: how do we move from $\succ$ to $\succsim$? Indeed, if $x\succsim z \succsim y$, there is the possibility in the end that $x\succsim y$, which is not really what we started from. Indeed, somewhere else I found a definition that starts from $\succ$ and ends with $\succ$ preserving the strict order.
Thanks in advance for any feedback.
If $x\in Z$, you can just choose $z= x$. Similarly, if $y \in Z$, just take $z = y$. Thus your definition can be rephrased as follows: for all $x,y \in X \setminus Z$ such that $x > y$, there exists some $z \in Z$ with $x \geq z \geq y$, ... or $x > z > y$ since now the two conditions are equivalent.