Solid torus

In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle.

[1] It is homeomorphic to the Cartesian product

of the disk and the circle,[2] endowed with the product topology.

A standard way to visualize a solid torus is as a toroid, embedded in 3-space.

However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus.

Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.

The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary.

The boundary is homeomorphic to

, the ordinary torus.

is contractible, the solid torus has the homotopy type of a circle,

[3] Therefore the fundamental group and homology groups are isomorphic to those of the circle:

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