1^∞, 0^0 and others on the real projective line

Hurkyl

Actually, 0^0 is relevant here. It's also a wonderful illustration of some of the relevant problems.

On the one hand, is not useful to give a value to 0^0. On the other hand, 0^0 is not only equal to 1, but it's not even a special case.

What changed from one side to the other is that ^ refers to the continuous exponentiation operator or its continuous extensions to things like the extended and projective real lines. ^, however, is being used to some sort of algebraic exponentiation operator; examples have domains including things like the base from any ring and exponent from the natural numbers. The exponentiation operation appearing in a power series is ^.