Let $A_n \subset X$ ($n \in \mathbb{N}$) be a sequence of subsets of a topological space $X$. Consider the following property:
(1) For any infinite set $I \subset \mathbb{N}$ the union $\bigcup_{n \in I} A_n$ is dense in $X$.
This can be equivalently rephrased as
(1') For any open set $U \subset X$ the set $\{ n \in \mathbb{N} \ : \ A_n \cap X = \emptyset \}$ is finite.
If $X$ is compact and metrizable and $d$ is a metric on $X$ then (1) is equivalent to
(2) For any $\varepsilon > 0$ there exists $n_0$ such that $A_n$ is $\varepsilon$-dense in $X$ for each $n \geq n_0$.
(Here, a set $A \subset X$ is $\varepsilon$-dense if for any $x \in X$ there exists $y \in A$ with $d(x,y) < \varepsilon$.)
This seems like a rather basic property so I would suppose it has an established name, but I can't seem to find one in the literature. Is there one?