On the metrizability of spaces with a sharp base (Q2782212)

Following \textit{B. Alleche}, \textit{A. V. Arkhangel'skij}, and \textit{J. Calbrix}, a \(T_1\)-space has a sharp base if it has a 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 \(x\) [Topology Appl. 100, No. 1, 23-38 (2000; Zbl 0935.54027)]. It is known that every topological space with a uniform base has a sharp base, and that every space with a sharp base is weakly developable (and therefore has a base of countable order). In this paper the following metrization theorems for spaces with a sharp base are obtained. NEWLINENEWLINENEWLINETheorem 1: Every pseudocompact Moore space with a sharp base is metrizable. NEWLINENEWLINENEWLINETheorem 2: If a pseudocompact regular space with a sharp base has a \(G^*_\delta\)-diagonal, then it is metrizable.NEWLINENEWLINENEWLINETheorem 3: Every regular, locally CCC, locally Baire space with a sharp base is metrizable.NEWLINENEWLINENEWLINETheorem 4: Every regular, \(\omega_1\)-compact space with a sharp base is metrizable.NEWLINENEWLINENEWLINETheorem 5: Every monotonically normal space with a sharp base is metrizable.NEWLINENEWLINENEWLINEHowever, the main contribution of this paper is the construction of a pseudocompact Tikhonov space \(X\) with a sharp base which has the following properties: (i) \(X\) is not metrizable; (ii) \(X\) has a weakly uniform base but no \(G^*_\delta\)-diagonal; (iii) \(X\times [0,1]\) does not have a sharp base; (iv) \(X\) has a weakly uniform base but is not compact. Whereas properties (i) and (iii) answer questions of Alleche, Arkhangel'skij, and Calbrix, properties (ii) and (iv) answer questions of \textit{R. W. Heath} and \textit{W. F. Lindgren} [Houston J. Math. 2, 85-90 (1976; Zbl 0318.54032)] respectively of \textit{S. A. Peregudov} [Quest. Answers Gen. Topology 17, No. 2, 153-155 (1999; Zbl 0936.54007)].NEWLINENEWLINEFor the entire collection see [Zbl 0983.00046].