Lindelöf degree of inverse limits and \(\sigma\)-products
The following results about the Lindelöf property of inverse limits and \(\sigma\)-products are known: I [\textit{K. Nagami}, Fundam. Math. 73, 261-270 (1972; Zbl 0226.54005)]. Assume that \(X_\infty = \varprojlim \{X_i, \pi_i^j\}\) is countably paracompact. If each \(\pi^j_i\) is open, each \(\pi_i\) is surjective and each \(X_i\) is a normal \(L(m)\) space, then \(X_\infty\) is a normal \(L(m)\) space. II [\textit{E. Michael}, Compos. Math. 23, 199-214 (1971; Zbl 0216.44304)]. The product \(\prod_{n < \omega} X_n\) is hereditarily Lindelöf iff for each \(n < \omega\) \(\prod_{i < n} X_i\) is hereditarily Lindelöf. III [the second named author, PhD Dissertation Sichuan Univ. 1990]. The \(\sigma\)-product of the spaces \(\{X_\lambda : \lambda \in \Lambda\}\) is Lindelöf iff for each \(c \in [\Lambda]^{< \omega} \prod_{\lambda \in c} X_\lambda\) is Lindelöf. The main results are the three theorems below. Theorem 1 is related to I and theorems 2 and 3 improve II and III, respectively. \(L(X)\) and \(hL(X)\) denote the Lindelöf degree and the hereditary Lindelöf degree, respectively. Theorem 1. Let \(X_\infty\) be the inverse limit of the inverse system \(\{X_\lambda, \pi^\mu_\lambda, \Lambda \}\). If each \(\pi_\lambda^\mu\) is open, \(\pi_\lambda\) surjective and \(X_\infty\) \(|\Lambda |\)-paracompact, then \(L(X_\infty) \leq|\Lambda|\cdot(\sup \{L(X_\lambda): \lambda\in\Lambda\})\). Theorem 2. Let \(X_\infty \) be the inverse limit of the inverse system \(\{X_\lambda, \pi_\lambda^\mu, \Lambda\}\). If each \(\pi_\lambda^\mu\) is open and each \(\pi_\lambda\) is surjective, then \(hL(X_\infty) \leq |\Lambda |\cdot (\sup \{hL (X_\lambda) : \lambda \in \Lambda\}) \). Theorem 3. Let \(X\) be the \(\sigma\)-product of the spaces \(\{X_\lambda:\lambda\in\Lambda\}\). Then \(L(X) = \sup \{L (\prod_{\lambda \in c} X_\lambda) : c \in [\Lambda]^{< \omega}\}\).
scientific article; zbMATH DE number 436764