I selected part of my mesh (the head of a character) and did a "Smart UV Project". For the rest of the mesh, I didn't do any UV operations. In the shader, I added a Texture Image and set an image. One copy of the image is now plastered across the head of the character, which is what I intended. But the image is also repeated all over the character, anywhere else where the material is applied.
From this I infer that for any face of the character where I didn't specify any UV mapping manually, one UV island is created. Is this correct?