Let $X$ be any topological space endowed with two topologies $\mathcal{T} \subset \mathcal{T}'$ (the later means that $\mathcal{T}'$ refines $\mathcal{T}$, that is every open subset $U \subset X$ with respect $\mathcal{T}$ (ie $U \in \mathcal{T})$ is already open with respect $\mathcal{T}'$.
Assume that $X$ is irreducible (or say more weaker connected;
irreducible implies connected) with respect $\mathcal{T}$.
Connected means that if there exist
two $U,V$ which are open & cloled for $\mathcal{T}$ and
$X= U \dot{\cup}V $ then either $U$ or $V$ is empty. And irreducible
that every two non empty open $U, V \in \mathcal{T}$ intersect properly:
$U \cap V \neq \emptyset$.
Question: Are there any interesting sufficient or neccessary conditions on $X, \mathcal{T}$ and $\mathcal{T}'$ known such that following holds
every open non empty $U \in \mathcal{T}$ is 'dense' in $X$ with respect finer topology $\mathcal{T}'$
What I mean by 'interesting'? Well, everything non trivial/boring like
$\mathcal{T}= \mathcal{T}'$ or $\mathcal{T}= \{X, \emptyset\}$ etc.
Note that this question generalizes this Question
where $X \subset \mathbb{C}^n$ is a complex irreducible variety, $\mathcal{T}$
the Zariski topology and $\mathcal{T}'$ analytical topology on $X$ induced
by $\mathbb{C}^n$