Is there a name for this property of a topology? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:05:17Z http://mathoverflow.net/feeds/question/18770 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18770/is-there-a-name-for-this-property-of-a-topology Is there a name for this property of a topology? Ketil Tveiten 2010-03-19T16:34:22Z 2010-03-20T15:01:23Z <p>This property seems like it should have a nice name, but I can't find one anywhere. Does anyone know a name for this?</p> <blockquote> <p>For each non-empty open set $U$, there exist proper open subsets $\{U_i\}_{i\in I}$ such that $U=\cup_i U_i$.</p> </blockquote> <p>I suppose this could also be formulated as each nonempty open set having an open cover of proper subsets, or being the colimit of its open subsets.</p> <p>(Also, apologies if this is something obvious I should have thought of.)</p> http://mathoverflow.net/questions/18770/is-there-a-name-for-this-property-of-a-topology/18775#18775 Answer by Joel David Hamkins for Is there a name for this property of a topology? Joel David Hamkins 2010-03-19T17:13:05Z 2010-03-19T17:32:48Z <p>In spaces where singleton points are closed, your property is equivalent to saying that the space has no isolated points. Or in other words, that it is <a href="http://en.wikipedia.org/wiki/Perfect_space" rel="nofollow">perfect</a>. </p> <p>Clearly, no space with an isolated point can have your property. Conversely, when singletons are closed, then you can subtract one point from any open set and thereby have a proper open subset. So if U has at least 2 points x,y, then U = U-{x} union U-{y}, giving an instance with I of size 2.</p> <p>However, your property does not imply that points are closed, since the space on reals R, where open sets have the form (-infty, a), has your property, but points are not closed in this space.</p> http://mathoverflow.net/questions/18770/is-there-a-name-for-this-property-of-a-topology/18827#18827 Answer by Andrej Bauer for Is there a name for this property of a topology? Andrej Bauer 2010-03-20T07:22:59Z 2010-03-20T07:22:59Z <p>Are you just saying that the topology is an <em>atomless</em> lattice? I'd call it "a space with atomless topology".</p> http://mathoverflow.net/questions/18770/is-there-a-name-for-this-property-of-a-topology/18834#18834 Answer by Undergrad for Is there a name for this property of a topology? Undergrad 2010-03-20T13:16:05Z 2010-03-20T13:16:05Z <p>Isn't this just the Base of the topology?</p> http://mathoverflow.net/questions/18770/is-there-a-name-for-this-property-of-a-topology/18842#18842 Answer by Alexei Fedotov for Is there a name for this property of a topology? Alexei Fedotov 2010-03-20T15:01:23Z 2010-03-20T15:01:23Z <p>Willie, this is ,clearly, saying that every local base can not be finite.</p> <p>I can not write comments that's why i am writing this like answer. Can I?</p>