complete vector spaces intuition I have been reading about vector spaces and want to get an understanding of completeness in a vector space.
My undestanding of compelteness comes from https://en.wikipedia.org/wiki/Complete_metric_space 
If there are other definitions of completeness I would really appreciate anyone pointing me to it.
Based on the above definition I have a few questions: 
For a space to be defined as complete, do we need a sequence of some sort? i.e., Completeness cannot be defined without a sequence? Further, this sequence needs to satisfy the closure property?
I am guessing a metric implies a set is ordered (or partially ordered)? Can anyone clarify if the set needs to be totally ordered for a metric or partially ordered?
Can I have a complete space without a metric?
 A: By definition, you can't leave a vector space by adding elements or performing multiplication with a scalar. (For $v,w \in V$ $v + w \in V$ and for $\lambda \in \mathbb R$, $\lambda v \in V$ if $V$ is a real vector space).
"Taking the limit" is a different operation and hence, if you have a sequence $v_n \in V$, you are not guaranteed that a limit exists, and if it exists that it lies in $V$. 
This is why complete vector spaces deserve a definition of their own. Cauchy sequences are essentially sequences that converge (see the note below for more details), and the notion of completeness makes sure that their limits are in $V$. In order to determine if a sequence is a Cauchy sequence, you need (at least) a metric.
Note: If you have a metric space $V$, you can define a Cauchy sequence in the usual way. Furthermore, you can define an equivalence relation on the set of Cauchy sequences in order to define the completion of $V$, see, e.g. here. This is why I said that Cauchy sequences 'have essentially a limit'.
A: Completeness is all about convergence of cauchy sequences inside the space . To study about the convergence to sequence in a space one must know the topological structure of that space so that one can define open sets, closed sets limit points and hence convergence of sequences with help of a metric.
