TITLE: Why is Kleene's notion of computability better than Banach-Mazur's?
QUESTION [15 upvotes]: In this post about the difference between the recursive and effective topos, Andrej Bauer said:

If you are looking for a deeper explanation, then perhaps it is fair to say that the Recursive Topos models computability a la Banach-Mazur (a map is computable if it takes computable sequences to computable sequences) and the Effective topos models computability a la Kleene (a map is computable if it is realized by a Turing machine). In many respects Kleene's notion of computability is better, but you'll have to ask another question to find out why :-)

So I'm asking: 
1) What is "computability a la Banach-Mazur"? I would guess it has something to do with Baire spaces and computable analysis, but I don't really know.
2) Why is Kleene's notion of computability better?

REPLY [3 votes]: Jacques asks about a "kernel" idea for the three counter-examples. I am not sure I can answer the question well. The most general result, namely the third one, is really a consequence of Friedberg's result. Peter Hertling's construction is an adaptation of Friedberg's result (although not an obvious one by any means), too. So the question is, what did Friedberg do? I must admit I cannot explain this succinctly, perhaps it's best if you have a look at Section 9 of

P. Hertling, "A Banach–Mazur computable but not Markov computable function on the computable real numbers", Annals of Pure and Applied Logic 132 (2005) 227–246.

Suffice it to say that the construction involves showing that a certain set of indices of B-M computable functionals is a $\Pi^0_4$-set, but the corresponding set of indices of Markov computable functionals is $\Sigma^0_4$-complete and contained in the first set, therefore the two sets differ. Now if there is anyone around who can explain in simple terms what $\Sigma^0_4$-complete sets are like, I would love to hear it.