TITLE: Why do we want complete spaces? We don't we just use closed spaces?
QUESTION [33 upvotes]: Why do we care about the notion of a space being complete? Why don't just consider closed spaces? If the space is closed we know that the limits of a sequence exist and are in the set which is a property that is obviously desirable.
So what is the benefit in introducing this weaker notion of complete spaces and dealing with these weaker types of sequences called Cauchy sequences, as opposed to just using closed spaces and the stricter convergent sequences? What are we allowed to do with complete spaces that wouldn't be possible with closed spaces? It would also be interesting to hear the historical motivation for complete spaces if anybody is aware of it.

REPLY [7 votes]: If the space is closed we know that the limits of a sequence exist 

That's... false. Try $0,1,0,1,\ldots$.
Did you mean that every sequence has a convergent subsequence? I hate to break this to you, but that's only true if the closed set is a subset of a complete space.
Or did you mean that every Cauchy sequence has a limit? Again...  that's only true in a complete space.
Completeness is important, you just have only been using examples of closed sets that are in complete spaces. And it sounds like the properties you "like" are actually properties of complete spaces. Bluntly you are confusing a lot of related concepts.

"Closed" is a topological property. "Complete" is a property of metric spaces only. $[0,1]$ is closed in $\mathbb R$ and $[0,1] \cap \mathbb Q$ is closed in $[0,1] \cap \mathbb Q$.
"Closed" only makes sense relative to a containing topological space. "Complete" is an intrinsic property.
"Limit points" can be defined just in terms of open sets and topology. Closed sets contain all their limit points in any topological set.
"Completeness" means if a sequence tries to converge, then it hs something to actually converge to. "Convergence" only makes sense in metric spaces. This is not true for $\mathbb Q$ where sequences might try to converge to $\sqrt 2$ for instance. (1, 1.4, 1.41, 1.414... is such a sequence).