Completion of topological spaces.

Question

Let be $Y \subset X$ be dense between rings with I-adic topology or between metric spaces. Can I conclude that the completion of $X$ is the same as the completion of $Y?$

I would think that it seems easy but pheraps technically so I wanted to know more about the problem.

Answer 1

As mentioned in comments, you don’t complete topological spaces, you complete metric spaces. There are examples of metric spaces which are homeomorphic, but where one is complete and the other is not, such as $\mathbb R$ and $(0,1)$ with their usual metrics.

You can embed $\mathbb Q$ in $(0,1)$ but the completion of $\mathbb Q$ is $\mathbb R$ and the completion of $(0,1)$ is $[0,1].$

You need the metric in $Y$ to be the restriction of the metric in $X$ for this to be true, not just a topological embedding.