Each time I open the color picker on Illustrator I have to change the color space to HSB. Since Illustrator always open with RGB by default, I was wondering whether there is a way to change the default color space option for the color picker.
You can not set the default color mode. It will always default back to the Document Color mode (RGB or CMYK). You can set it, but you'll quickly find after clicking a few objects, it'll revert back to the default color mode.
There's no way to permanently set it to a specific color model. This fix has been requested for years.