Hyperreal Fields
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 M is a maximal ideal in C(X). Then the factor algebra A = C(X)/M is a totally ordered field F containing the reals. If F strictly contains R then M is called a hyperreal ideal (terminology due to Hewitt (1948)) and F a hyperreal field. Note that no assumption is being made that the cardinality of F is greater than R; it can in fact have the same cardinality.
An important special case is where the topology on X is the discrete topology; in this case X can be identified with a cardinal number κ and C(X) with the real algebra of functions from κ to R. The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory.
Read more about this topic: Hyperreal Number
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“Genius is the naturalist or geographer of the supersensible regions, and draws their map; and, by acquainting us with new fields of activity, cools our affection for the old. These are at once accepted as the reality, of which the world we have conversed with is the show.”
—Ralph Waldo Emerson (1803–1882)