Is there a locally compact, $\omega_1$-compact, not $\sigma$-countably compact space of size $\aleph_1$? - MathOverflow most recent 30 from mathoverflow.net 2019-10-20T17:53:58Z https://mathoverflow.net/feeds/question/228284 https://creativecommons.org/licenses/by-sa/4.0/rdf https://mathoverflow.net/q/228284 15 Is there a locally compact, $\omega_1$-compact, not $\sigma$-countably compact space of size $\aleph_1$? Peter Nyikos https://mathoverflow.net/users/85198 2016-01-13T00:07:56Z 2016-09-14T20:56:23Z <p>There are old ZFC examples due to Eric van Douwen that satisfy all the properties in the title, except that they are of cardinality $2^{\aleph_0}$, so the answer to the title question is YES if the Continuum Hypothesis (CH) is assumed, but I am asking for an example which does not use any axiom beyond the ZFC axioms. </p> <p>It would be especially nice if it were first countable, like Eric's examples, and normal, like one of his examples. A Souslin tree with the interval topology qualifies, but the existence of Souslin trees is ZFC-independent. I have been able to weaken CH to "stick" [which says that there is a family of $\aleph_1$ countable subsets of an uncountable set, such that every uncountable subset contains a member of the family] and I also have an example if $\mathfrak b = \aleph_1$ but no ZFC example.</p> <p>It would also be interesting to know whether the existence of an example implies the existence of a first countable example. There are lots of first countable examples under CH, and the "stick" example I have in mind is also first countable, as is my $\mathfrak b = \aleph_1$ example. For sure, there is a scattered example if there is one at all: the "Kunen line" qualifies under CH, while if CH is negated, then we use the fact that every crowded (= dense-in-itself) compact Hausdorff space is of cardinality at least $2^{\aleph_0}$.</p> https://mathoverflow.net/questions/228284/is-there-a-locally-compact-omega-1-compact-not-sigma-countably-compact-s/249901#249901 6 Answer by Peter Nyikos for Is there a locally compact, $\omega_1$-compact, not $\sigma$-countably compact space of size $\aleph_1$? Peter Nyikos https://mathoverflow.net/users/85198 2016-09-14T20:08:31Z 2016-09-14T20:56:23Z <p>Lyubomyr Zdomskyy has solved this problem. He has shown:</p> <p>Theorem. If P-Ideal Dichotomy (PID) holds and $\mathfrak p &gt; \aleph_1$, then every locally compact, $\omega_1$-compact space of cardinality $\aleph_1$ is $\sigma$-countably compact. </p> <p>I've forgotten my password, so I am using Google to "Sign up".</p>