Given a metrizable space X, let $AM(X)$ be the space of continuous bounded admissible metrics on X, which is endowed with the sup-metric. In this paper, we shall investigate the Borel complexity and the complete metrizability of $AM(X)$ and show that a separable metrizable space X is σ-compact if and only if $AM(X)$ is completely metrizable.
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Katsuhisa Koshino (2024) studied this question.
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