In mathematics, particularly topology, the tube lemma, also called Wallace's theorem, is a useful tool in order to prove that the product of finitely many compact spaces is compact.

Statement

The lemma uses the following terminology:

• are topological spaces is the product space, endowed with the product topology, a slice.

• A tube.

Using the concept of closed maps is closed.

Examples and properties

in the product topology, that is the Euclidean plane is a tube). This shows that the compactness assumption is essential.

is compact as follows:

be an open cover is compact, being homeomorphic By the tube lemma, there is an open set.

3. By part 2 and induction, one can show that the finite product of compact spaces is compact.

4. The tube lemma cannot be used to prove the Tychonoff theorem, which generalizes the above to infinite products.

Proof

are respectively open, which completes the proof.

See also

References

• (See Chapter 8, Lemma 8.9)