How to actually invert the fuzzy selection?

[rich2005]

I think this came up before, especially for small selected parts of an image.
You should experiment with the threshold value for the fuzzy select tool.
or use a path to make the selection, Where the path crosses a pixel (sub-pixel) pixel transparency is adjusted.

   

(1) Black selected, selection inverted
(2) Selection cut.
(3) Inverted selection, path from selection.
(4) Selection from path, selection cut.