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With the operations of pointwise addition and pointwise
multiplication, is a commutative topological ring with
unit, the unit being the constant function with value 1. It is
a
famous theorem of Gelfand and
Kolmogoroff [18] that the
ring structure by itself determines the topological structure on
provided is compact. They proved that if and are
compact and and are isomorphic as rings then and
are homeomorphic. For details,
see also Dugundji [17, Theorem XIII.6.5].
(The proof in [17] makes use of the
topology of pointwise convergence.)
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 Arhangelski [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 .

** Next:** Topological equivalence of function
** Up:** Preliminaries
** Previous:** - and -equivalence