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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