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 <math>(l)</math> may be studied in isolation —in which case the ambient space of <math>l</math> is <math>l</math>, or it may be studied as an object embedded in 2-dimensionalEuclidean space <math>(\mathbb{R}^2)</math>—in which case the ambient space of <math>l</math> is <math>\mathbb{R}^2</math>, or as an object embedded in 2-dimensional hyperbolic space <math>(\mathbb{H}^2)</math>—in which case the ambient space of <math>l</math> is <math>\mathbb{H}^2</math>. To see why this makes a difference, consider the statement "Parallel lines never intersect." This is true if the ambient space is <math>\mathbb{R}^2</math>, but false if the ambient space is <math>\mathbb{H}^2</math>, because the geometric properties of <math>\mathbb{R}^2</math> are different from the geometric properties of <math>\mathbb{H}^2</math>. All spaces are subsets of their ambient space.
