This is getting peripheral to the main point. AS far as I recall, I thought of T as a test object. As for epsilon-delta, Bishop required that delta be prescribed as a constructible function of epsilon in order that a function be continuous. He required that the convergence be uniform on every closed interval, so that this function on a closed interval was independent of the points in the interval. Michael On Wed, 21 Jan 2009, Charles Wells wrote:
Calculus teachers do something similar when they make an epsilon-delta proof into a game: The opponent picks an epsilon (the test object) and you have to come up with a delta. There is one big difference between epsilon-delta proofs and limits. To show that something is a limit you have to find, for each test object, the unique arrow specified by the definition of limit. Thus you are producing a function (indeed, a bijection). The delta for a given epsilon is not unique, and so there is no natural function giving a delta for each epsilon. I am pretty sure this makes epsilon-delta proofs harder for non-talented students than proving something is a limit. I know some calculus teachers talk about there being a function that takes epsilon to delta, but I suspect it is a mistake to bring that up.
Charles Wells