- Superreal number
The superreal numbers are an extension of the
real numbers, similar to the surreal numbersor hyperreal numbers, but comprising a more inclusive category than either one.
Suppose X is a
Tychonoff space, also called a T3.5 space, and C(X) is the algebra of continuous real-valued functions on X. Suppose P is a prime idealin C(X). Then the factor algebra A = C(X)/P is by definition an integral domain which is a real algebra and which can be seen to be totally ordered. The quotient field F of A is a superreal field if F strictly contains the real numbers , so that F is not order isomorphic to , though they may be isomorphic as fields.
If the prime ideal P is a maximal ideal, then F is a field of
The terminology is due to Dales and Woodin.
* H. Garth Dales and W. Hugh Woodin: "Super-Real Fields", Clarendon Press, 1996.
* L. Gillman and M. Jerison: "Rings of Continuous Functions", Van Nostrand, 1960.
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