The \(d\)-rank of a topological space
For a \(T_0\) topological space \(X\) let \(D(X)\) denote the set of \(\sim\)-equivalence classes of non-empty upward directed subspaces under the specialisation order, where \(A\sim B\) provided \(A\cap U\not=\emptyset\Leftrightarrow B\cap U\not=\emptyset\) for every open set \(U\subset X\). For each open \(U\subset X\) let \(U^*\) consist of those equivalence classes \([S]\) such that \(S\cap U\not=\emptyset\): the set of all such \(U^*\) forms a topology on \(D(X)\). \(X\) naturally embeds in \(D(X)\). Then a transfinite sequence \(\langle D_\alpha(X)\rangle\) may be formed by setting \(D_0(X)=X\), \(D_{\alpha+1}(X)=D(D_\alpha(X))\) and \(D_\lambda(X)=\lim_{\to\lambda}\langle D_\alpha(X)\rangle\) (\(\lambda>0\) a limit ordinal). The \textit{\(d\)-rank} of \(X\) is the least ordinal \(\alpha\) such that \(D_\alpha(X)=D_{\alpha+1}(X)\). It is shown that for any ordinal \(\alpha\) there is an irreducible \(T_0\) space whose \(d\)-rank is \(\alpha\).
scientific article; zbMATH DE number 6822481
scientific article