- Superreal number
The

**superreal numbers**are an extension of thereal numbers , similar to thesurreal numbers orhyperreal number s, but comprising a more inclusive category than either one.**Formal Definition**Suppose X is a

Tychonoff space , also called a T_{3.5}space, and C(X) is the algebra of continuous real-valued functions on X. Suppose P is aprime ideal in 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 $Bbb\{R\}$, so that F is not order isomorphic to $Bbb\{R\}$, though they may be isomorphic as fields.If the prime ideal P is a maximal ideal, then F is a field of

hyperreal number s.The terminology is due to Dales and Woodin.

**References*** 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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