Key points are not available for this paper at this time.
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 -compact if and only if AM (X) is completely metrizable.
Katsuhisa Koshino (Tue,) studied this question.