I opened up a defective hotplate and found a bunch of cotton-like insulation
material inside. I put them into a plastic bag but a lot of short fibers
(less than 1cm long) fall out of the "cotton" which I vacuumed up.

Here's a photo of it:
http://www.smugmug.com/gallery/20384895_GsHfRJ

I'm guessing it's fiberglass or asbestos. How can I tell? And is any of
these two material harmful to be around? If I fix the hotplate, should I put
this "cotton" back, or should I use something else?

The heating wires is inside a sheet of mylar-like material and the "cotton"
insulation is used to press the sheet tight against the top of the hotplate.