Independent subbases and non-redundant codings of separable metrizable spaces
An independent subbase of a space \(X\) is a subbase \(S= \{S_{n,i}:n<\omega,i<2\}\) for the topology on \(X\) such that for all \(n<\omega\), \(S_{n,0}\cap S_{n,1}=\emptyset\) and for every \(\sigma\in {}^{<\omega}2\), the set of all functions from a non-negative integer into \(2=\{0,1\}\), \(\bigcap_{k<n}S_{k,\sigma(k)}\not=\emptyset\). The authors prove that a separable metric space \(X\) is dense-in-itself if and only if \(X\) has an independent subbase. This answers a question raised by the second author in [Topol. Proc. 28, No.~2, 673--687 (2004; Zbl 1125.54018)]. The authors say that an independent subbase \(S= \{S_{n,i}:n<\omega,i<2\}\) has dimension \(m\) provided \(\{X\setminus(S_{n,0}\cup S_{n,1}):n<\omega\}\) has order at most \(m-1\), where the order of a family of sets \(\mathcal A\) is the largest non-negative integer \(m\) such that \(\mathcal A\) contains \(m+1\) sets with non-empty intersection. Let \(\dim(X)\) denote the covering dimension of a separable metric space \(X\). The authors prove that a separable metric space \(X\) is dense in itself and has \(\dim(X)\leq m\) if and only if \(X\) has an independent subbase of dimension \(m\). These results are proved using \(\mathbb T^\omega\)-codes, where \(\mathbb T=\{0,1,\perp\}\), where \(\perp\) is called the bottom element. Such codes arise in computability theory.
scientific article; zbMATH DE number 5835571
scientific article