![cover image](https://wikiwandv2-19431.kxcdn.com/_next/image?url=https://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Euclidian_and_non_euclidian_geometry.png/640px-Euclidian_and_non_euclidian_geometry.png&w=640&q=50)
Ambient space (mathematics)
The space surrounding an object / From Wikipedia, the free encyclopedia
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 may be studied in isolation —in which case the ambient space of
is
, or it may be studied as an object embedded in 2-dimensional Euclidean space
—in which case the ambient space of
is
, or as an object embedded in 2-dimensional hyperbolic space
—in which case the ambient space of
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.
![]() | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (May 2024) |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Euclidian_and_non_euclidian_geometry.png/640px-Euclidian_and_non_euclidian_geometry.png)