 Completely metrizable space

In mathematics, a completely metrizable space (metrically topologically complete space) is a topological space ("X", "T") for which there exists at least one metric "d" on "X" such that ("X", "d") is a complete metric space and "d" induces the topology "T".The term topologically complete space is employed by some authors as a synonym for "completely metrizable space", but sometimes also used for other classes of topological spaces, like completely uniformizable spaces or Čech-complete spaces.