Name for a kind of topological property? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:04:35Z http://mathoverflow.net/feeds/question/36194 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36194/name-for-a-kind-of-topological-property Name for a kind of topological property? Jeff Strom 2010-08-20T14:36:59Z 2010-08-20T23:06:47Z <p>What should I call a property (P) of (open) subspaces of a space \$X\$ such that:</p> <ol> <li><p>If \$U\$ satisfies (P), then so does every open subset \$V\subset U\$</p></li> <li><p>If {\$U_i\$} is a pairwise disjoint collection of sets satisfying (P), then \$\bigcup_i U_i\$ satisfies (P). (Unable to make braces?)</p></li> </ol> <p>My understanding is that if (P) satisfies condition 1, then (P) is called a hereditary property. </p> <p>CLARIFICATION: My main question is really: is there existing terminology for such a property? </p> <p>I will, however be happy to consider suggestions on the secondary question: if not, then what should I call it?</p> http://mathoverflow.net/questions/36194/name-for-a-kind-of-topological-property/36235#36235 Answer by Pietro Majer for Name for a kind of topological property? Pietro Majer 2010-08-20T23:06:47Z 2010-08-20T23:06:47Z <p>I would call such a property <em>hereditary and (completely) additive on open sets</em>. If you want to specify a cardinality constraint on the index set I, then adverbs like finitely/countably (or sigma) may be useful.</p>