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 —in which case the ambient space of l is, or as an object embedded in 2-dimensional hyperbolic space —in which case the ambient space of l 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.