Ambient space (mathematics)

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

Three examples of different geometries: Euclidean, elliptical, and hyperbolic