Closest star system to Alpha Centauri?
The sun is the nearest star to Alpha Centauri (unless you count Proxima Centauri, which is really part of the same system).
There is a very small and dim pair of brown dwarfs, called Luhman 16 that are closer, at about 3.6 light years from Alpha Centauri. Brown dwarfs are not true stars, but they do glow from their own heat. They were only discovered in 2011. It is possible that there are other very dim objects, but these would not be stars either in the strict sense.
To add for context on the distance of Luhman 16: the Sun is around 4.2 light-years from Alpha Centauri. (Thanks!)
It's a bit confusing when you say the Sun is the nearest star, but then say there is a pair closer. I think what you are saying is that these are not stars in the strict sense, but since that attaches to the "other very dim objects" it's not entirely clear that it applies to these two stars as well.
I don't think I said that there are a pair of stars closer, since I said "brown dwarfs", but I've added a short edit.
@JamesK, I tried Googling this question but had little success (it's relatively easy to find distances to/from our Solar System, but not between other systems). For posterity (and out of curiosity), how did you find out the distances you mentioned between Luhman 16 and Alpha Centauri (for example)?
For the distances between stars I used the recons data http://www.recons.org/ (I found a version that has the positional data in xyz format, which simplifies things. Recons only has stars, I think I found luhman 16 in a web search, but i forget the magic words but the result was on Wikipedia
If you have XYZ galactic coordinates for two stars, wouldn't the distance between them just be SQRT((X(1) - X(2))^2 + (Y(1)-Y(2))^2 + (Z(1)-Z(2))^2)).