Weak bases and metrizability (Q2778267)

Let \(X\) be a topological space and consider the following conditions on a function \(g:\omega\times X\to {\mathcal P}(X)\):NEWLINENEWLINENEWLINE(i) \(x\in\bigcap \{g(n,x) \mid n<\omega\}\) for all \(x\in X\);NEWLINENEWLINENEWLINE(ii) \(g(n+1,x) \subset g(n,x)\) for all \(n< \omega\) and \(x\in X\);NEWLINENEWLINENEWLINE(iii) a subset \(U\) of \(X\) is open if and only if for every \(x\in U\) there is an \(n<\omega\) such that \(g(n,x)\) is contained in \(U\);NEWLINENEWLINENEWLINE(iv) \(\text{cl} A\subset \bigcup\{g(n,x) \mid x\in A\}\) for each subset \(A\) of \(X\) and for each \(n<\omega\);NEWLINENEWLINENEWLINE(v) if for each \(n< \omega\), \(y_n\in g(n,x_n)\) and the sequence \((y_n)\) converges to \(p\) in \(X\), then \(p\) is a cluster point of the sequence \((x_n)\);NEWLINENEWLINENEWLINE(vi) if \(x\in g(n, y_n)\), \(x_n\in g(n,y_n)\), \(y_n\in g(n,x_n)\), and \(y\in g(n,x)\) for all \(n< \omega)\), then the sequence \((x_n)\) converges to \(x\).NEWLINENEWLINENEWLINEIt is shown that a topological space \(X\) is metrizable if and only if it has a function \(g:\omega \times X\to {\mathcal P}(X)\) satisfying either (i)--(v), or (i)--(iv) and (vi).