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

Abstract

Given a metrizable space XX, let AM(X)AM\left(X) be the space of continuous bounded admissible metrics on XX, which is endowed with the sup-metric. In this article, we shall investigate the Borel complexity and the complete metrizability of AM(X)AM\left(X) and show that a separable metrizable space XX is σ\sigma -compact if and only if AM(X)AM\left(X) is completely metrizable.

Keywords

• admissible metric
• sup-metric
• absolute borel class
• completely metrizable
• primary 54c35
• secondary 54e35
• 54h05