Estimations of small transfinite dimension in separable metrizable spaces (Q1848087)

Transfinite versions of familiar dimension functions come in many flavours. Some are straightforward generalizations, such as trind, defined by \(\operatorname {trind}X\leq\alpha\) if there is a base \(\mathcal B\) for \(X\) such that \(\operatorname {trind}\operatorname {Fr}B<\alpha\) for all \(B\in\mathcal B\). Others are more intricate such as Henderson's \(D\)-dimension: \(D(X)\) is the smallest ordinal \(\alpha\) for which \(X\) has a closed cover \(\{A_\beta:\beta\leq\lambda(\alpha)\}\) with 1) \(\bigcup_{\beta=\delta}^{\lambda(\alpha)}A_\beta\) is always closed, 2) \(\max\{\beta:x\in A_\beta\}\) always exists, and 3) \(\dim A_\beta<\infty\) and \(\dim A_{\lambda(\alpha)}=n(\alpha)\). Here \(\lambda(\alpha)\) and \(n(\alpha)\) are the unique limit and finite ordinals respectively with \(\alpha=\lambda(\alpha)+n(\alpha)\). The authors show how trind can be estimated in terms of \(D\), as follows: if \(D(X)=\alpha\geq\omega_0\) then \(\operatorname {trind}X\leq \lambda(\alpha)+m+1\), where \(m\) satisfies \(2^m\geq n(\alpha)+1\) if \(X\) is compact and \(2^m\geq n(\alpha)+2\) in general. This greatly improves older estimates such as those in [\textit{L. A. Luxemburg}, Pac. J. Math. 93, 339-386 (1981; Zbl 0397.54048)] and implies that very often \(\operatorname {trind}X<D(X)\). The proofs rest on earlier results by the first-named author from [Fundam. Math. 144, 95-117 (1994; Zbl 0809.54027) and Fundam. Math. 162, 91-98 (1999; Zbl 0938.54031)]; these are also used to show that \(\operatorname {trind}(X\times Y)<\operatorname {trind}X+\operatorname {ind} Y\) more often than not when \(Y\) is finite-dimensional.