Ambient space (mathematics)

{{short description|The space surrounding an object}}

{{distinguish-redirect|Ambient space|Ambient space (psychology)}}

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File:Euclidian and non euclidian geometry.png, elliptical, and hyperbolic]]

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