TITLE: preservation of completeness under homeomorphism
QUESTION [12 upvotes]: Does homeomorphic metric spaces preserves completeness?I mean two metric space which are homeomorphic and one of them is complete$\Rightarrow$ another one is also complete?

REPLY [4 votes]: The answer is not always. If a space is homeomorphic to a complete metric space we say that it is completely metrizable, namely there exists a metric which is complete and generates our given topology. We know that a space is completely metrizable if and only if it is a $G_\delta$ set (intersection of countably many open sets) in every complete metric space embedding it, in particular in its metric completion.
However if the metric space is compact then it is always complete. To see this note that a Cauchy sequence in a compact metric space is an infinite set and therefore has a limit point, however being a Cauchy sequence it can have at most one limit point - the limit of the sequence. Therefore every Cauchy sequence is convergent and thus the metric is complete.