A Niemytzky-Tychonoff theorem for all topological spaces

The classical Niemytzky-Tychonoff theorem characterises compactness of a metrisable topological space by means of the completeness of all of the metrics inducing the topology. Motivated by results of Kopperman and Flagg to the effect that every topological space is metrisable, as long as metrisability is suitably modified to allow the metric to take values more general than real numbers, we show that the Niemytzky-Tychonoff theorem remains true under this broader notion of metrisability, thus obtaining a metric characterisation of compactness valid for all topological spaces.