Q:

Partition into open sets and their boundary

Let $\tau$ be a topology on $X$. 
Let $\omega \subseteq \tau$ be a set of pairwise disjoint, open subsets of $X$.
Let $Y := \bigcup\limits_{Z \in \omega} Z$.
Question: If $X = Y \cup \partial Y$, does $\omega$ have a name? I.e., which terms are typically used for such $\omega$?

A:

As Brian said: $\omega$ is a maximal family of (non-empty) pairwise disjoint open sets in $X$. Maximal is meant in the sense of inclusion. 
For suppose $O$ is an non-empty open set in $X$, and suppose $O$ is disjoint from all members of $\omega$ (so that $\omega \cup \{O\}$ is a strictly larger pairwise disjoint family). Then $O \cap Y = \emptyset$, but this means that any $x \in O$ is neither in $Y$ nor in $\partial Y$ which cannot be. So $\omega$ is maximal.
The reverse is also clear: if $\omega$ is maximal, and $x \notin Y$, then let $O$ be any open set that contains $x$. Then $O$ must intersect $Y$, as otherwise $O$ is disjoint from any $Z \in \omega$ so that $\omega \cup \{O\}$ is a larger pairwise disjoint family of non-empty sets, which cannot be. And $O$ intersects $X \setminus Y$ in $x$ already, so $x \in \partial Y$ and so $Y \cup \partial Y = X$.   

