Is there a locally compact, $\omega_1$-compact, not $\sigma$-countably compact space of size $\aleph_1$? - MathOverflowmost recent 30 from mathoverflow.net2019-10-20T17:53:58Zhttps://mathoverflow.net/feeds/question/228284https://creativecommons.org/licenses/by-sa/4.0/rdfhttps://mathoverflow.net/q/22828415Is there a locally compact, $\omega_1$-compact, not $\sigma$-countably compact space of size $\aleph_1$?Peter Nyikoshttps://mathoverflow.net/users/851982016-01-13T00:07:56Z2016-09-14T20:56:23Z
<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#2499016Answer by Peter Nyikos for Is there a locally compact, $\omega_1$-compact, not $\sigma$-countably compact space of size $\aleph_1$?Peter Nyikoshttps://mathoverflow.net/users/851982016-09-14T20:08:31Z2016-09-14T20:56:23Z<p>Lyubomyr Zdomskyy has solved this problem. He has shown:</p>
<p>Theorem. If P-Ideal Dichotomy (PID) holds and $\mathfrak p > \aleph_1$,
then every locally compact, $\omega_1$-compact space of cardinality $\aleph_1$
is $\sigma$-countably compact. </p>
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