Cam anyone provide me the proof of:


  that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is connected.



Question: Can you designate your craft, perform, or profession skill even if you have no ranks in it?
Answer: Not only can you, you have to. You never roll a &ldquo;Craft check&rdquo; or &ldquo;Perform check,&rdquo; those skills don&rsquo;t exist. Rather, they are categories of skills, like &ldquo;Craft (basketweaving)&rdquo; or &ldquo;Perform (underwater basketweaving).&rdquo; That&rsquo;s still true even if you have no ranks, and it&rsquo;s relevant for things like feats or instruments, which may need to be compatible with the chosen Craft or Perform skill.

Craft and Perform may be used without ranks, so if you have high Intelligence or high Charisma, and/or other bonuses to those skills, you could even be good at them without any training (e.g. ranks). It&rsquo;d be hard to keep up with someone who is actually training it, though.

Profession&rsquo;s much the same, but unless you are trained, you cannot roll the specific Profession skill at all (instead, you just get a flat 1 sp/day as unskilled labor). So having ranks in Profession (woven-basket critic) doesn&rsquo;t let you roll for Profession (woven-basket recycler) checks; they are separate skills, and you have no ranks in the latter. But you still have to make the distinction between the two.