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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.

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?

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.

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.

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 $\leftbrace U_i \rightbrace$ is a pairwise disjoint collection of sets satisfying (P), then $\bigcup_i U_i$ satisfies (P).

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

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.