Weak developments and metrization
The authors call a topological space \(X\) weakly developable if it has a sequence \(({\mathcal U}_n)_{n<\omega}\) of open covers such that for every sequence \((U_n)_{n <\omega}\) with \(U_n\in {\mathcal U}_n\) and for every \(x\in\bigcap \{ U_n\mid n<\omega\}\) the sequence \((\bigcap\{U_i\mid i\leq n\})_{n <\omega}\) is a local base at \(x\). Moreover, a topological space is said to have a sharp base if it has an open base \({\mathcal B}\) such that for every sequence \((B_n)_{n< \omega}\) of distinct members of \({\mathcal B}\) and for every \(x\in\bigcap \{B_n\mid n< \omega\}\) the sequence \((\bigcap \{B_i\mid i\leq n\})_{n <\omega}\) is a local base at \(x\). Obviously, every developable topological space is weakly developable, and every topological space with a uniform base has a sharp base. It is shown that every topological space with a sharp base is weakly developable. An interesting example is constructed which proves that spaces with a sharp base need not be developable. Moreover, a new characterization of spaces having a base of countable order is obtained from which it follows that every weakly developable topological space has a base of countable order. It is shown that \(\omega_1\) has a base of countable order but is not weakly developable. Additionally, it is shown that Čech-complete Moore spaces as well as pseudocompact Moore spaces have a weak \(k\)-development.
scientific article; zbMATH DE number 1395070