Q-sets and Bing's tangent-disk Moore-space interface

Work in $\mathrm{Z}\mathrm{F}\mathrm{C}$. All Euclidean distances are the usual ones on ${\mathbb{R}}^2$.

Q-sets. A subset $E\subseteq \mathbb{R}$ is a Q-set when $E$ is uncountable and every subset $A\subseteq E$ is relatively $G_{\delta }$: there are open sets $V_n\subseteq \mathbb{R}$, $n\in \mathbb{N}$, with $A=E\cap {\bigcap }_{n\in \mathbb{N}}{V}_n\mathrm{.}$ (Equivalently, replacing $V_n$ by ${\bigcap }_{k\le n}{V}_k$, the sets may be assumed decreasing.) Nothing here asserts that a Q-set exists; existence is proved from Martin's axiom together with $\mathrm{\neg }\mathrm{C}\mathrm{H}$ in the later item Martin's axiom produces an uncountable Q-set.

The tangent-disk space. Let $E\subseteq \mathbb{R}$ and put $Z(E):=(\mathbb{R}\times (0,\mathrm{\infty }))\cup (E\times \{0\}),$ with the subspace convention that $E\times \{0\}$ is the axis part and $P:=\mathbb{R}\times (0,\mathrm{\infty })$ the open part. For $n\in \mathbb{N}$ with $n\ge 1$ and $p\in Z(E)$ define $U(p,n):=\{\begin{array}{ll}\{z\in P:\mathrm{dist}(p,z)<1\mathrm{/}n\}& p\in P,\\ \{p\}\cup \{z\in P:\mathrm{dist}((p_1,1\mathrm{/}n),z)<1\mathrm{/}n\}& p=(p_1,0)\in E\times \{0\}\mathrm{.}\end{array}$ The tangent-disk topology (the Moore plane topology restricted to $Z(E)$) is the topology generated by the basis $\mathcal{B}:=\{U(p,n):p\in Z(E),n\ge 1\}$ (Basis and subbasis for a topology, and the topology generated by a family of sets, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). This is a basis: every point belongs to its own sets $U(p,n)$. At an axis point in an intersection, both sets are tangent disks based at that same point, and the disk of radius $1\mathrm{/}\max (n,m)$ lies in both. At a point of $P$, their intersection is Euclidean open in $P$ and contains a sufficiently small $U(p,k)$. Therefore unions of these sets form a topology, since intersections are unions of such smaller members. In this topology a point $p=(a,0)$ of the axis part has the tangent disk $U(p,n)$ as basic neighbourhoods, an open disk whose boundary is internally tangent to the axis at $(a,0)$, together with that point; the axis part is closed in $Z(E)$ and each $U(p,n)$ meets it in $\{p\}$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

The pair $(E,Z(E))$ is the interface recorded here: development, separability, normality and nonmetrizability of $Z(E)$ for uncountable Q-sets $E$ are proved in Bing's Q-set space is a normal nonmetrizable Moore space.

• Why the tangent disks. The closure of the Euclidean disk $B((a,1\mathrm{/}n),1\mathrm{/}n)$ touches the axis exactly at $(a,0)$, so $U((a,0),n)\cap (\mathbb{R}\times \{0\})=\{(a,0)\}$; the axis points are therefore isolated within the axis part, and the axis part is a closed discrete subspace of $Z(E)$. That discreteness is what turns an uncountable Q-set into a nonmetrizable Moore space, and the Q-set property is exactly what restores normality.

• The open part is Euclidean. $P$ carries its usual metric topology (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, Open ball, closed ball and sphere in a metric space), and it is dense in $Z(E)$.