Is one-to-one correspondence the same as bijection? Or is it a bijection that is everywhere defined ?
 A: I believe there is a subtle difference between a bijection and a so called one-to-one-correspondence, especially in popular math treatments of Cantor's set theory. The difference being that a one-to-one-correspondence is typically drawn with bi-directional arrows - so the distinction between the domain and range is not really meaningful - while a bijection is drawn with one directional arrows (viz. vectors) from elements of the domain pointing to elements of the co-domain. Strictly speaking a bijection is a mapping with extra features, so one directional arrows are appropriate.
After establishing a bijection does exist between two sets, one can immediately use the inverse relation to get the other direction of each arrow, though we are now implicitly using a theorem that the inverse relation of a bijection is also a bijection.
On a side note, unfortunately a one-to-one function or mapping, sans the word correspondence, has come to be known as an injection, so the extra word 'correspondence' seems to add the surjectivity aspect.
In conclusion a one-to-one-correspondence isn't a function in the strict mathematical sense of what a function is, it's a term that has come to have a meaning of its own outside of function terminology. Specifically a one-to-one correspondence is a matching up or pairing up of the elements of two sets, such that no element of the first set is unpaired with an element of the second set and vice a versa . Diagrammatically it is easiest to picture this pairing up using bi-directional arrows between the elements of the two sets.
I am afraid I may just be splitting hairs here.
