2
$\begingroup$

What are examples for maximality statements that cannot be proved using Zorn's Lemma?

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ I think the title and the actual question don't fit together. $\endgroup$
    – Stefan Kohl ♦
    Commented Apr 17, 2015 at 15:10
  • 1
    $\begingroup$ "existence of ultrafilters" (or BPI) is an answer to the question in the title, but not to the actual question. $\endgroup$
    – Goldstern
    Commented Apr 17, 2015 at 16:02

1 Answer 1

Reset to default
5
$\begingroup$

A statement I like very much is:

Any compact topology is contained in a maximal compact topology.

A direct application of Zorn's Lemma doesn't work, as the following argument shows. Consider $\omega$ as ground set, and set $\tau_n := \cal{P}(\{0,\ldots,n\}) \cup \{\omega\}$ for $n\in\omega$. Then $\{\tau_n:n\in \omega\}$ is a chain of compact topologies, but its supremum is the discrete topology on $\omega$, which is not compact.

Despite failure of a direct application of $\mathsf{ZL}$, Martin Maria Kovar proved here that every compact topology is contained in a maximal compact topology, settling an open question that was asked by D. E. Cameron in 1977.

Improve this answer
$\endgroup$