$X \cong Y$, $X$ complete $\implies Y$ complete?

Question

Let $\cong$ denote the homeomorphic notation.
Let $X,Y$ be metric spaces, and let $X \cong Y$. If $X$ is a complete metric space does it imply $Y$ is also complete.

Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be (temporarily) closed. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. — Lord_Farin, May 14 '13 at 07:03
Example of Homeomorphism Between Complete and Incomplete Metric Spaces, preservation of completeness under homeomorphism, Is Completeness intrinsic to a space? — Martin Sleziak, May 14 '13 at 07:38

score 5 · Answer 1 · answered May 14 '13 at 05:39

5

Hint: $(0,1)$ in the standard metric is incomplete, and $\Bbb R$ is complete.

answered May 14 '13 at 05:39

Asaf Karagila

1 Answers1