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

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

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?

hereditaryis generally reserved for properties which are inherited byallsubspaces, not just open ones. To be honest, I would not introduce a name for such a thing, but only a shorthand («Let us, for briefness, callexcellenta property such that ... and ...», and then talk about «excellent properties»; if the concept catches up, this makes it more probable that you get immortalized with «Stromian property» or something!) – Mariano Suárez-Alvarez♦ Aug 20 '10 at 14:51excellentorgood(unless you are really trying to get it named after you), it maybe better to make a definition to the effect that a property (P) satisfying conditions (1) and (2) are said to be in class S. – Willie Wong Aug 20 '10 at 15:39