 Completely metrizable space

A completely metrizable space is a topological space that can be given a metric (or distance function) such that the topology induced by this metric is the same as the original topology of the space, and furthermore, the metric is complete. To break this down: 1. **Topological Space**: This is a set of points, along with a collection of open sets that satisfy certain axioms (like closure under unions and finite intersections).