I got confused with these two concepts when consider the set $\Omega$ of real valued continuous functions defined on $[0, 1]$. By definition, $\Omega$ is certainly closed since every set is a closed subset of itself. However, $\Omega$ is not complete. One counter example is the function sequence $(f_n) = x^{n}$. Pointwisely $(f_n)$ converges to the function $f = I_{\{0<x\leq 1\}} + 0\times I_{\{x=0\}}$, which is not continuous.
Now reconsider the above example. By definition a set is closed if it contains all its limit points. However, $\Omega$ does not seem to satisfy this criterion. That is, $\Omega$ is not closed, which contradicts the previous statement. I am wondering where I understand wrongly. Thank you!
You're right that a closed set must contain all of its limit points. But one must be careful: the definition states that the limit points must be in the space. So in your example, the limit of the sequence $x^n$ is indeed not in the space, but that does not contradict the set being closed.
For an example of what I mean, consider the metric space $X=[0,1)$ with the standard Euclidean metric. The subset $[1/2,1)$ is closed as a subset of $X$; the point 1 is not in $X$, and hence does not count as a limit point of $[1/2,1)$ in $X$.
What this does show is that the space is not universally closed, which means that it isn't closed in any larger metric space. So if you add in the limit of that sequence to the space and make the space larger, then your $\Omega$ won't be closed as a subspace of that larger metric space.