Topologists are interested in the *classification* of
topological spaces. Classifying topological spaces up to
homeomorphism
or homotopy type etc., is the
ultimate goal for a topologist.
We are interested here
in classifying spaces
up to homeomorphism or linear-homeomorphism type of their
function spaces . In its full generality this program
would be much too complicated and the
complete picture is
presently beyond our reach. But for function spaces of low Borel complexity
some definitive results are known, and it is our aim to discuss
them here.

We say that spaces and are *-equivalent*
provided that and are *linearly*
homeomorphic. Notation:
.

Homeomorphic spaces are obviously -equivalent. But the
converse need not be true.
Let
and
. Then evidently,
and are not homeomorphic. However, they are
-equivalent. Indeed, define
by

Then is a linear homeomorphism.

We say that and are *-equivalent*
provided that are are homeomorphic as
topological spaces.
Notation: .

Even for simple spaces it is in general difficult to decide whether they are - or -equivalent. By Bessaga and Peczynski [8] there are countable compact spaces and for which the Banach spaces and are not linearly homeomorphic. An application of the Closed Graph Theorem shows that if and are linearly homeomorphic then so are and (the same linear map does the job in both cases). So the examples of Bessaga and Peczynski are not -equivalent. This suggests the question whether they are -equivalent. We will come back to this below.

Arhangelski [1] proved that if
is compact and
is linearly homeomorphic to then is compact.
As a consequence,
and
are not
linearly homeomorphic. But they *are*
homeomorphic, as was shown by Gul'ko and
Khmyleva [19].

Results in the same spirit were obtained by various authors. Pestov [28] proved that if and are linearly homeomorphic then and have the same dimension. So and are not linearly homeomorphic. Observe that by the famous result of Miljutin [24], all Banach spaces with uncountable and compact metrizable are linearly homeomorphic. Hence and are linearly homeomorphic, but and are not. For another result in the same spirit, see Baars, de Groot and Pelant [7].