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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.

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?

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    $\begingroup$ I would say that hereditary is generally reserved for properties which are inherited by all subspaces, 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, call excellent a 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!) $\endgroup$
    – Mariano Suárez-Álvarez
    Aug 20, 2010 at 14:51
  • $\begingroup$ Rather than using pedestrian adjectives like excellent or good (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. $\endgroup$
    – Willie Wong
    Aug 20, 2010 at 15:39
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    $\begingroup$ I find adjectives more readable. Compare "Take a good property and assume..." with "Take a property of class S and assume..." $\endgroup$
    – Andrea Ferretti
    Aug 20, 2010 at 15:56
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    $\begingroup$ I should have said this before: if you are going to pick an adjective, please oh please let it be not «admissible»! $\endgroup$
    – Mariano Suárez-Álvarez
    Aug 20, 2010 at 19:45
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    $\begingroup$ @Mariano. I was thinking of "regular." $\endgroup$
    – Jeff Strom
    Aug 20, 2010 at 21:01

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I would call such a property hereditary and (completely) additive on open sets. If you want to specify a cardinality constraint on the index set I, then adverbs like finitely/countably (or sigma) may be useful.

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