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
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 Pe
czynski [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
Pe
czynski 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].