some questions on Lindelöf property - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:34:50Z http://mathoverflow.net/feeds/question/84726 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84726/some-questions-on-lindelof-property some questions on Lindelöf property Paul 2012-01-02T09:37:03Z 2012-01-04T21:45:03Z <p>I have several questions on Lindelöf property.</p> <p>If every point countable open cover of $X$ has a countable subcover (<strong>Condition A</strong>), does $X$ have Lindelöf property? How far is having <strong>Condition A</strong> from Lindelöf property?</p> <p><strong>A space $X$ is called $\omega_1$-Lindelöf if every $\omega_1$-sized open cover of $X$ contains a countable subcover.</strong></p> <p>Can every $\omega_1$-Lindelöf space with <strong>Condition A</strong> be Lindelöf?</p> <p><strong>A space $X$ is called discretely Lindelöf if the closure of every discrete subspace of $X$ is Lindelöf.</strong></p> <p>Can every discretely Lindelöf space with <strong>Condition A</strong> be Lindelöf?</p> http://mathoverflow.net/questions/84726/some-questions-on-lindelof-property/84728#84728 Answer by Ricky Demer for some questions on Lindelöf property Ricky Demer 2012-01-02T10:34:13Z 2012-01-02T10:34:13Z <p>$X = \langle \omega_1,2^{\omega_1} \rangle$</p> <p>The set of singleton subsets of $\omega_1$ is a point countable $\omega_1$-sized <br> open cover of $X$ that does not have a countable subcover. <br><br><br> I don't know the answers to your other two questions.</p> http://mathoverflow.net/questions/84726/some-questions-on-lindelof-property/84905#84905 Answer by Henno Brandsma for some questions on Lindelöf property Henno Brandsma 2012-01-04T21:45:03Z 2012-01-04T21:45:03Z <p>A space is Lindelöf iff it satisfies condition A and is metaLindelöf (every open cover has a point-countable refinement), so one could say the difference is metaLindelöfness.</p>