Any space contains a dense left-separated subspace.
Given any space $X$, form a (possibly transfinite) sequence of points by the following induction. As long as (the range of) the sequence you've built so far isn't dense in $X$, choose arbitrarily a point not in its closure, and append the chosen point to your sequence. Stop only when your sequence has become dense in $X$. Every initial segment of (the range of) your sequence is closed because, after it was formed, you only added points outside its closure.
My question is this:
If $X$ is a compact space, what toplogical properties can its dense left-separated subspace have?
For example, it is Tychonoff. Thanks advance!