## Proof That the Cylinder, Torus and Cube Are Compact Spaces

Theorem

The cylinder, torus and cube are compact spaces.

Proof

The cube is homemorphic to the product spaceinwith the usual Euclidean topology.

is compact inwith the usual Euclidean topology. Use Tchonoff's Theorem then the product spaceand the cube are compact.

The torus and cylinder are homeomorphic to the product spacesandrespectively, hence both are compact by Tychonoff's Theorem.