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Proof that a subspace $A$ of a complete metric space $X$ is complete . . . The fact that complete and closed are different concepts is because you can have closed sets in incomplete metric spaces However, this theorem shows us that in any complete metric space, completeness and closure of a set is the same thing
Is every complete metric space closed? - Mathematics Stack Exchange No Every metric induces what is called a topology on the underlying set, and the notions of open and closed sets in metric spaces generalize to notions of open and closed sets in topological spaces If you’ve not encountered topological spaces at all, my comment won’t make much sense, and I’ll substitute a weaker statement: every metric space is a closed subset of itself, complete or not
Metric space is complete - Mathematics Stack Exchange The definition of a complete metric space is one where all Cauchy sequences converge So, take an arbitrary Cauchy sequence $\ {x_n\}$ and show that for all $\epsilon>0$ $\exists N\in\mathbb {N}$ such that $\forall n,m > N$, $|x_n-x_m|<\epsilon$ implies that $\ {x_n\}$ converges What you have written there is the definition of a Cauchy sequence
Motivation behind the definition of complete metric space What is motivation behind the definition of a complete metric space? Intuitively,a complete metric is complete if they are no points missing from it How does the definition of completeness (in t
Formal proof for completion of metric space 4 In this pdf, a method for constructing a completion for a given metric space $ (X, d)$ It is very similar (and infact equivalent) to the one in this question
What does it mean for a Metric to be complete? 1 A metric space is complete if every Cauchy sequence converges in the space A metric is complete if the metric space it generates is complete