Well, I can't explicitly construct one, but one can talk about cardinality. The cardinality of [0, 1] = the cardinality of the real numbers (which is uncountably infinite). The cardinality of (0, 1) = [0, 1] - {0, 1} is also equal to the cardinality of [0, 1] by cardinal arithmetic. Similarly, the cardinality of (0, 1] = [0, 1] - {0} is also the cardinality of [0, 1]. Now, by definition of cardinality, two sets have the same cardinality if there exists a bijection between the two.
Fair enough. I've never heard cardinality be defined that way, but it certainly makes sense. And now that I'm thinking about it, I guess we want there to be a bijection intuitively anyway. We'll let f-inverse(1)=1/2, and let the inverse function map the rest of (0,1) evenly on either side of 1/2. If that makes any sense...
AAAHHH JESUS CHRIST MAKE IT STOP :(