Locally compact, 0-dimensional, pseudocompact space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:47:24Z http://mathoverflow.net/feeds/question/95857 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95857/locally-compact-0-dimensional-pseudocompact-space Locally compact, 0-dimensional, pseudocompact space Fred Dashiell 2012-05-03T12:46:56Z 2012-05-03T15:58:58Z <p>Is a 0-dimensional, locally compact and pseudocompact space $X$ necessarily strongly 0-dimensional? I.e., must $\beta X$ be 0-dimensional?</p> <p>It is known that a 0-dimensional locally compact space which is also paracompact must be strongly 0-dimensional (Engelking, 1989, p. 362). But the answer to a recent question posted here points out that $\omega_1$ is locally compact and pseudocompact but not paracompact, so an approach attempting to use that fact will not answer the present question.</p> http://mathoverflow.net/questions/95857/locally-compact-0-dimensional-pseudocompact-space/95882#95882 Answer by Ramiro de la Vega for Locally compact, 0-dimensional, pseudocompact space Ramiro de la Vega 2012-05-03T15:53:41Z 2012-05-03T15:53:41Z <p>An infinite collection $\mathcal{A}$ of infinite subsets of $\mathbb{N}$ is said to be almost disjoint (AD) if $A\cap B$ is finite whenever $A,B \in \mathcal{A}$ with $A \neq B$. If the family is maximal with respect to this property, then it is called a MAD family.</p> <p>Given an AD family $\mathcal{A}$, there is a well known way (introduced by Mrowka in "On completely regular spaces", Fund.Math.,1954) to construct a topological space $\Psi(\mathcal{A})$. This space has the following properties:</p> <p>1) For any AD family, $\Psi(\mathcal{A})$ is 0-dimensional, locally compact and first countable.</p> <p>2) $\Psi(\mathcal{A})$ is pseudocompact if and only if $\mathcal{A}$ is a MAD family.</p> <p>Teresawa (in "Spaces N∪R need not be strongly 0-dimensional", Bull. Acad. Polon. Sci. Sér. Sci. Math. Astonom. Phys., 1977) proved that there is a MAD family $\mathcal{A}$ for which $\Psi(\mathcal{A})$ is not strongly 0-dimensional. This provides a counterexample to your question.</p> http://mathoverflow.net/questions/95857/locally-compact-0-dimensional-pseudocompact-space/95884#95884 Answer by KP Hart for Locally compact, 0-dimensional, pseudocompact space KP Hart 2012-05-03T15:58:58Z 2012-05-03T15:58:58Z <p>The spaces in <a href="http://mathoverflow.net/questions/93719/0-dimensional-locally-compact-space/93762#93762" rel="nofollow">this answer</a> are pseudocompact.</p>