Sum theorems for the strong small transfinite dimension (Q1375725)

The authors obtain the following theorems, where \(\text{sind } X\) stands for the strong small transfinite dimension introduced by \textit{P. Borst} [Infinite-dimension theory (Part II), manuscript (1981)]. Theorem 1. Let \(X\) be a strongly hereditarily normal space. If \(\mathcal C = \{ C_i\}_{i\in I}\) is a locally finite closed cover of \(X\) such that for each \(i\in I \text{ sind} (C_i)\leq\xi\), then \(\text{sind} (X)\leq \xi\). Theorem 2. Let \(X\) be a regular space. If \(X\) admits an open cover \(\mathcal R =\{G_i\}_{i\in I}\) such that \(\text{sind} (G_i)\leq\xi\) for each \(i\in I\), it holds \(\text{sind } X\leq\xi\). Two corollaries are also obtained.