## Sometimes a maximality principle is true even if Zorn’s Lemma fails

A famous and straightforward result in general topology says that any compact $T_2$ space is

• maximally compact: any finer topology isn’t compact any more;
• minimally $T_2$: any coarser topoology isn’t $T_2$ any more.

A natural pair of questions that arises from this is the following: Is every compact topology contained in a maximally compact one? (Dually: does every Hausdorff topology contain a minimal Hausdorff topology?).

This short post is only concerned with the former of the two questions. Unfortunately, a direct application of Zorn’s Lemma doesn’t lead us anywhere. Let $\mathbb{N}$ denote the set of positive integers. We define a chain of compact topologies on $\mathbb{N}$ by setting

$\tau_n = \mathcal{P}(\{1,\ldots,n\}) \cup \{\mathbb{N}\}$ for every $n\in \mathbb{N}$. (By $\mathcal{P}(.)$ we denote the power set.)

Now every $\tau_n$ is compact, but the topology generated by the union of this chain is the discrete topology on $\mathbb{N}$, which is not compact. So we cannot apply Zorn’s Lemma for this question.

It turns out that even if we can’t apply ZL, the statement that every compact topology is contained in a maximally compact one is true, which answered an old question. The result can be seen in this paper by Martin Maria Kovar.