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