# Admissible Representations in Type-2 Theory of Effectivity

Matthias Schroeder Department of Computer Science University of Hagen (FernUniversität in Hagen) 4pm 16 September 2003 Room 2511, JCMB, King's Buildings

The property of admissibility of representations plays an important
role in Type-2 Theory of Effectivity (TTE). TTE yields a computational
framework for uncountable sets *X* which are equipped with a
natural approximation structure.
Such an approximation structure may be given by a topology or by a
convergence relation on *X*. The basic idea of TTE is to represent
the objects of *X* by elements of the Baire space and to perform
the real computation on these names rather than on the objects themselves.
The corresponding partial surjection *d* onto *X*
is called a representation of *X*.

For functions between two represented spaces *(X,d)* and
*(Y,c)* there are two notions of continuity:
The first one, sequential continuity, is given by the approximation
structure; the second one, relative continuity, is defined relatively
to representations by saying that *f* is relatively continuous
iff there is a continuous function on the Baire space realizing *f*
w.r.t. the involved representations. Relative continuity describes the
finitary aspects of computations w.r.t. the considered representations.
Admissibility of representations is defined in such a way that equivalence
of both continuity notions is guaranted.

The category of sequential topological spaces having an admissible representation turns out to be cartesian closed. Moreover, it is equal to the category of (topological) quotients of countably-based spaces and also to the category PQ of \Omega-projecting quotients of countably-based spaces. Hence a wide range of spaces used in Analysis can be appropriately covered by TTE.