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72 Jazz Bass pickups

"Open" is a break in the coil. You can sometimes still hear sound because the windings are capacitively coupled to each other because they are so close to one another.

If the insulation is worn, that's a short, not an open coil. Shorts often happen when the wire corrodes where it touches the magnets, and then it shorts to the magnets. Sometimes one wind will short to another. The pickup will still work, but might sound dull.
 
The pickup still works but the sound is very thin and weak. The thing that has been bugging me is the tone working like a volume?

That's an open coil.

Is it better to just have both the pickups redone if one is bad?

No, just have the bad one rewound. It's not hard to make them match.