In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object along with the object itself. For example, a 1-dimensionalline may be studied in isolation —in which case the ambient space of is , or it may be studied as an object embedded in 2-dimensionalEuclidean space—in which case the ambient space of is , or as an object embedded in 2-dimensional hyperbolic space—in which case the ambient space of is . To see why this makes a difference, consider the statement "Parallel lines never intersect." This is true if the ambient space is , but false if the ambient space is , because the geometric properties of are different from the geometric properties of . All spaces are subsets of their ambient space.
Schilders, W. H. A.; ter Maten, E. J. W.; Ciarlet, Philippe G. (2005). Numerical Methods in Electromagnetics. Vol. Special Volume. Elsevier. pp. 120ff. ISBN0-444-51375-2.