Symmetric and tufted assignments of neighbourhoods and metrization

We identify the concept of a tufted assignment of neighbourhoods and with it strengthen a remarkable theorem of Nagata to have: Every metrizable space has a metric with respect to which balls of equal radii constitute a tufted and symmetric assignment of neighbourhoods. We also have: The availability on a T-3-space of a basic sequence of tufted and symmetric assignments of neighbourhoods is (necessary and) sufficient for metrizability. Hausdorff spaces are paracompact if and only if open covers have refinements in the form of tufted and symmetric assignments of neighbourhoods. Moore spaces X are metrizable if (and only if) given any open cover W, there is such a sequence <{U-n(X): X epsilon X}> of tufted and symmetric assignment of neighbourhoods that, for every X epsilon X, U-n(x) subset of St(x, W) for some n. T-3-spaces are strongly metrizable if and only if on them there are basic sequences of symmetric, point-finite assignments of neighbourhoods. (C) 2008 Elsevier B.V. All rights reserved.