Showing that every sequential KC space is Katětov-KC

Question

We recall that if P is a topological property, then a space $\langle X, \tau\rangle$ is said to be minimal P (respectively maximal $P$) if $\langle X, \tau\rangle$ has property P but no topology on $X$ which is strictly smaller (respectively, strictly larger) than $\tau$ has P. A space $\langle X,\tau\rangle$ is said to be Katětov-P if there is a topology $\sigma\subseteq\tau$ such that $\langle X,\sigma\rangle$ is minimal P. In particular, a KC-space $\langle X,\tau\rangle$ is said to be Katětov-KC if there is a minimal KC-topology $\sigma\subseteq\tau.$.

We know that :

1. A second countable minimal KC-space is compact Hausdorff.

2. A first countable KC-space is minimal KC iff it is compact Hausdorff.

3. A sequential minimal KC-space is compact.

But why is every sequential KC space Katětov-KC?

Answer 1

See your own question here. This is based on the paper here. In that paper this corollary that a sequential KC space is Katětov KC is a corollary of theorem 2.4.

The proof is based on the following simple fact:

Let $(X, \tau)$ be KC and compact. Then it is minimally KC. Proof: suppose $(X,\sigma)$ is such that $\sigma \subset \tau$ and $(X, \sigma)$ is KC. Then let $O \subset X$ be in $\tau$, so $X \setminus O$ is $\tau$-closed, hence $\tau$-compact (as $(X,\tau)$ is compact). But then $X \setminus O$ is also $\sigma$-compact (as $\sigma \subset \tau$; use the definition of compactness) and so $X \setminus O$ is $\sigma$-closed (as $(X,\sigma)$ is KC). But then $O$ is $\sigma$-open, i.e. $O \in \sigma$, and we have shown that $\tau \subset \sigma$, hence $\sigma = \tau$. So any topology on $X$ that is coarser than $\tau$ and also KC, must equal $\tau$, showing that $\tau$ is indeed minimally KC.

Now suppose $(X, \tau)$ is sequential KC. Then the construction from the linked question (also used in theorem 2.4 in the paper) and its proof in the paper, show there is a topology $(X, \sigma)$ that is KC and compact and such that $\sigma \subset \tau$. This $\sigma$ is thus (by the fact I just proved) minimally KC, and this $\sigma$ witnesses that $(X,\tau)$ is Katětov KC by definition.

So it's a corollary of the proof of theorem 2.4, not directly of the theorem itself.