For noncompact spaces , the algebraic structure of
is,
in general, not strong enough to determine the topology of
.
For consider the spaces
and
. Then
clearly
and
are isomorphic as rings, but
and
are not homeomorphic.
For arbitrary spaces there is a result in the same spirit though.
Nagata [27]proved that and
are topologically isomorphic as topological rings if and
only if
and
are homeomorphic. That we deal with real
valued functions is essential in this result. It
was shown in Arhangel
ski
[3, page 12]
that the ring of all continuous functions
,
endowed with the topology of pointwise convergence, does not
always determine the topological type of
.