I opened a PSD in Illustrator. The file has a path in it, but when given a stroke, it joins the ends of the path, which I do not want. I do not want a closed stroke, instead I want an open stroke.
This happens because your path is being imported from Photoshop as a Compound Shape. Strokes, or any appearance attributes, are applied to the whole shape.
To stop this from happening you need to release the compound shape from the Pathfinder panel's options dropdown:
You can find out more about Compound Shapes here: