(adsbygoogle = window.adsbygoogle || []).push({}); [topology] "The metric topology is the coarsest that makes the metric continuous"

1. The problem statement, all variables and given/known data

Let (X,d) be a metric space. Show that the topology on X induced by the metric d is the coarsest topology on X such that [itex]d: X \times X \to \mathbb R[/itex] is continuous (for the product topology on [itex]X \times X[/itex]).

2. Relevant equations

N.A.

3. The attempt at a solution

I can prove that d is continuous, but I'm having trouble proving that the topology is the coarsest. Let V be a subset of R, then denote [itex]U := d^{-1}(V)[/itex] and suppose U is open. I want to prove that V is open, but I'm not sure how.

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# [topology] The metric topology is the coarsest that makes the metric continuous

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