Moore's clopen-generated topology

Retain the binary colouring $c(\alpha ,\beta )=o(\alpha ,\beta )<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$ from The Moore colouring realizes finite binary patterns. For $\alpha <{\omega }_1$, put

$W_{\alpha }=\{\alpha \}\cup \{\beta :\alpha <\beta <{\omega }_1\text{ and }c(\alpha ,\beta )=1\}\mathrm{.}$

For $X\subseteq {\omega }_1$, let $\tau [X]$ be the topology on $X$ generated by declaring $W_{\xi }\cap X$ clopen for every $\xi \in X$. Equivalently, finite intersections of sets $W_{\xi }\cap X$ and $X\setminus W_{\xi }$, with $\xi \in X$, form a clopen base. The empty intersection is $X$; when $X$ is empty this gives its unique topology.

There is a useful product representation with no extra generators. Let $\mathbb{T}$ be the unit circle and define

$e_X:X⟶\{-1,1{\}}^{{\omega }_1}\subseteq {\mathbb{T}}^{{\omega }_1}$

by

$e_X(x)(\xi )=\{\begin{array}{ll}-1,& \xi \in X\text{ and }x\in W_{\xi },\\ 1,& \text{otherwise.}\end{array}$

For $\xi \in X$, the inverse images of the two coordinate values are $W_{\xi }\cap X$ and its complement; for $\xi \mathrm{\notin }X$, the coordinate is constant. Thus the product-induced topology is exactly $\tau [X]$. If $x<y$ are in $X$, then $x\mathrm{\notin }{W}_y$ while $y\in W_y$, so the $y$-coordinate separates their images. Hence $e_X$ is injective and is an embedding onto its image.

The family $\{W_{\xi }\cap X:\xi \in X\}$ is point-countable and point-separating: if $x\in W_{\xi }$, then $\xi \le x$, and the countable ordinal $x+1$ has only countably many members. The displayed clopen base and point separation make every $(X,\tau [X])$ zero-dimensional Hausdorff, hence regular under the library convention.

Coordinates outside $X$ were deliberately padded by the constant $1$. Allowing membership in $W_{\xi }$ at such a coordinate would add subbasic sets that Moore did not put into $\tau [X]$.