Does Niemytzki plane have a $\sigma$-locally finite $k$-network?

Question

Niemytzki plane (a.k.a. Moore plane) is the Euclidean open half-plane together with the points on the edge with tangent open discs (and the point itself) as base neigbourhoods (cf. pi-Base, Wikipedia). It is a famous example of a Tychonoff but not normal space.

A $k$-network is a family $\mathcal N$ of subsets of $X$ such that for every compact set $K$ and open set $U$ in $X$ with $K\subseteq U$, there exists a finite $\mathcal{N}^* \subseteq \mathcal{N}$ with $K \subseteq \bigcup\mathcal{N}^* \subseteq U$. The family $\mathcal N$ is $\sigma$-locally finite if it is a countable union of locally finite families (see pi-Base, Does the Everywhere doubled line have a $\sigma$-locally finite $k$-network?).

A $k$-network is something intermediate between a base for the topology and a network. A family $\mathcal N$ of subsets of $X$ is called a network if every open set is the union of a subfamily of $\mathcal N$.

A $\sigma$-locally finite network for the Niemytzki plane can be constructed out of a countable basis of topology on the half-plane and the family of all singletons on the edge.

In this self-answered question I determine whether a $k$-network can be found.

Answer 1

To show that Niemytzki plane does not have a $\sigma$-locally finite $k$-network the proof from https://math.stackexchange.com/q/5034751 can be adapted:

Proof: Denote the half-plane by $H:=\mathbb R{\times}(0,+\infty)$. Assume $\mathcal N$ is a $\sigma$-locally finite $k$-network. Then $\mathcal D:=\{ E\cap H: E\in\mathcal N\}$ is a $\sigma$-locally finite family with respect to $H$. Since the half-plane is Lindelöf, any locally finite family has to be at most countable and so is $\mathcal D$.

The family $\mathcal D'$ consisting of finite unions of the members of $\mathcal D$ is countable as well.

For every $x\in\mathbb R$ consider a compact set $K_x:=\{x\}\times[0,1]$ and its open neighborhood $U_x:=\{(x,0)\}\cup B_e((x,1),1)$ (tangent disc with radius $K_x$). Then there exists finite $\mathcal N_x\subset \mathcal N$ such that $K_x\subset \bigcup \mathcal N_x \subset U_x$. Especially $A_x:=H\cap \bigcup\mathcal N_x$ is an element of $\mathcal D'$. Since $A_x\subset U_x$ we have $\overline{A_x}\setminus H =\{(x,0)\}$, hence the subfamily $\{A_x:x\in\mathbb R\}$ of $\mathcal D'$ is not countable. A contradiction. $\square$

Similarity of the proofs almost suggests that there is a common property of Niemytzki plane and Everywhere doubled line preventing the existence of a $\sigma$-locally finite $k$-network.