As a ward against any confusion—because I expect at least one person will make this mistake after reading your comment—it should be noted that while as a topological space it’s metrizable, the resulting metric on the reals would necessarily have pretty different properties (it would be the same topologically, but not uniformly, e.g.).
As a ward against any confusion—because I expect at least one person will make this mistake after reading your comment—it should be noted that while as a topological space it’s metrizable, the resulting metric on the reals would necessarily have pretty different properties (it would be the same topologically, but not uniformly, e.g.).