What should I call a property (P) of (open) subspaces of a space $X$ such that:

1.  If  $U$  satisfies (P), then so does every open subset $V\subset U$

2.  If {$U_i$} is a pairwise disjoint collection of sets satisfying (P), then 
$\bigcup_i U_i$ satisfies (P).  (Unable to make braces?)


My understanding is that if (P) satisfies condition 1, then (P) is called a
hereditary property.  


CLARIFICATION:  My main question is really:  is there existing terminology for such 
a property?  

I will, however be happy to consider suggestions on the secondary question: if not, then what should I call it?