On the Borel complexity and the complete metrizability of spaces of metrics

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 $\sigma$-compact if and only if $AM(X)$ is completely metrizable.