What is the keyspace of ROT-13?
I think it's better to define the  keyspace of an unkeyed function as having one element.

Some advantages:


Computing the key size as $log_2 1$ correctly tells you it's a 0 bit key
For encryption you pick one key from the set. You can pick an element of a single element set, but can't pick from an empty set
Functions with multiple inputs are often defined to take the Cartesian product of their inputs. For example for encryption this could be $M \times K \to C$ where $M$ is the space of messages, $K$ the key space, and $C$ the ciphertext space. One again, this only works correctly if you use a single element set as key space.