Definition:Open Sigma-Discrete Set of Subsets

Definition
Let $T = \struct {S, \tau}$ be a topological space.

Let $\UU$ be a set of subsets of $S$.

Then $\UU$ is open $\sigma$-discrete :
 * $(1) \quad \UU \subseteq \tau$, that is, for all $U \in \UU: U$ is open in $T$
 * $(2) \quad \UU$ is $\sigma$-discrete.