Ordered fundamental spaces and nice refinements

A fundamental space is a Hausdorff space $(X,\mathcal{T})$ of cardinality ${\mathrm{\aleph }}_1$ that is zero-dimensional, separable, first countable, and ${\mathrm{\aleph }}_1$-dense (every nonempty open set has size ${\mathrm{\aleph }}_1$), together with strictly decreasing clopen local bases

$X=W_0^x⊋W_1^x⊋W_2^x⊋\cdots ,{\bigcap }_{n<\omega }{W}_n^x=\{x\}\mathrm{.}$

It is ordered when $X={\omega }_1$ and $\omega$ is dense. The adjective does not say that $W_n^{\alpha }\subseteq \alpha$: indeed $W_0^{\alpha }={\omega }_1$.

Fix an ordered fundamental space. Suppose $V_n^{\xi }\subseteq W_n^{\xi }$ and $V_n^{\xi }\searrow \{\xi \}$. For $\alpha \le {\omega }_1$, set

$R_n^{\xi }(\alpha )=\{\begin{array}{ll}{V}_n^{\xi },& \xi <\alpha ,\\ W_n^{\xi },& \alpha \le \xi ,\end{array}$

and let ${\mathring{\mathcal{T}}}_{\alpha }$ be the topology generated by all these $R_n^{\xi }(\alpha )$; put $\tilde{\mathcal{T}}={\mathring{\mathcal{T}}}_{{\omega }_1}$. A nice refinement requires:

• $\max V_n^{\alpha }=\alpha$, and $V_n^{\alpha }$ is closed while $V_n^{\alpha }\cap \alpha$ is open in ${\mathring{\mathcal{T}}}_{\alpha }$;
• $V_n^{\alpha }=\{\alpha \}$ for $\alpha <\omega$;
• for $\omega \le \alpha <{\omega }_1$, there are nonempty, pairwise-disjoint, compact clopen sets $H_{\mathrm{\ell }}^{\alpha }\subseteq \alpha \cap (W_{\mathrm{\ell }}^{\alpha }\setminus W_{\mathrm{\ell }+1}^{\alpha })$ such that $V_m^{\alpha }=\{\alpha \}\cup {\bigcup }_{\mathrm{\ell }\ge m}{H}_{\mathrm{\ell }}^{\alpha },$ and $\{V_m^{\alpha }:m<\omega \}$ is a clopen local base at $\alpha$ in $\tilde{\mathcal{T}}$.

For the model-guided version, fix a continuous suitable chain $⟨M_{\alpha }:\alpha <{\omega }_1⟩$ of countable elementary submodels, put the global assignment $(\alpha ,n)\mapsto W_n^{\alpha }$ in $M_1$, and require the earlier $V$-sequence and final topology below $\alpha$ to belong to $M_{\alpha +1}$. On $\alpha +1$, choose a metric $d_{\alpha }$ for ${\mathring{\mathcal{T}}}_{\alpha }\upharpoonright (\alpha +1)$ whose ring $(W_{\mathrm{\ell }}^{\alpha }\setminus W_{\mathrm{\ell }+1}^{\alpha })\cap (\alpha +1)$ lies at radial value $2^{-\mathrm{\ell }}$ from $\alpha$ and has internal diameter at most $2^{-\mathrm{\ell }-3}$ in $d_{\alpha }$.

For $\omega \le \alpha <{\omega }_1$, if $S\in M_{\alpha +1}$ is countable and $S\subseteq K(\alpha ,\tilde{\mathcal{T}}\upharpoonright \alpha )$, define, for $\nu \le \omega$,

${\mathrm{\Gamma }}_{\alpha }(S,\nu )=\{K\in S:(\mathrm{\forall }\mathrm{\ell }<\nu )K\cap (W_{\mathrm{\ell }}^{\alpha }\setminus W_{\mathrm{\ell }+1}^{\alpha })\subseteq H_{\mathrm{\ell }}^{\alpha }\}\mathrm{.}$

Here $K(Y)$ denotes the nonempty compact subsets of $Y$, with the Vietoris topology generated by the subbasic conditions $K\subseteq U$ and $K\cap U\mathrm{\ne }\mathrm{\varnothing }$ for open $U\subseteq Y$. Its standard basic neighbourhoods are $N(U_0,\dots ,U_r)=\{K:K\subseteq {\bigcup }_{i\le r}{U}_i\text{ and }K\cap U_i\mathrm{\ne }\mathrm{\varnothing }\text{ for every }i\le r\}$; in a zero-dimensional space the $U_i$ may be taken nonempty, pairwise-disjoint and clopen. Distances between compacta use the Hausdorff metric induced by $d_{\alpha }$. For the same range $\omega \le \alpha <{\omega }_1$, the approximation condition $({\star }_{\alpha })$ is

$\mathrm{\forall }S[\begin{array}{c}S\in M_{\alpha +1},\mathrm{\mid }S\mathrm{\mid }\le {\mathrm{\aleph }}_0,\\ S\subseteq K(\alpha ,\tilde{\mathcal{T}}\upharpoonright \alpha )\end{array}⟹{\mathrm{\exists }}^{\mathrm{\infty }}\nu <\omega \mathrm{\forall }K\in {\mathrm{\Gamma }}_{\alpha }(S,\nu )\mathrm{\exists }J\in {\mathrm{\Gamma }}_{\alpha }(S,\omega )d_{\alpha }(K,J)\le 2^{-\nu }]\mathrm{.}$

The condition permits $S$ and either $\mathrm{\Gamma }$-set to be empty; its implication then has the literal vacuous meaning. The model chain, compatible metrics, and recursive selections are ZFC choices and account for the declared AC dependency. They are auxiliary construction data, not part of the raw definition of a fundamental space.