Ambient space (mathematics)

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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.

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