Tube lemma

In mathematics, particularly topology, the tube lemma, also called Wallace's theorem, is a useful tool in order to prove that the finite product of compact spaces is compact. The lemma uses the following terminology: If and are topological spaces and is the product space, endowed with the product topology, a slice in is a set of the form for . A tube in is a subset of the form where is an open subset of . It contains all the slices for . Using the concept of closed maps, this can be rephrased concisely as follows: if is any topological space and a compact space, then the projection map is closed.

1. Consider in the product topology, that is the Euclidean plane, and the open set The open set contains but contains no tube, so in this case the tube lemma fails. Indeed, if is a tube containing and contained in must be a subset of for all which means contradicting the fact that is open in (because is a tube). This shows that the compactness assumption is essential.
2. The tube lemma can be used to prove that if and are compact spaces, then is compact as follows: Let be an open cover of . For each , cover the slice by finitely many elements of (this is possible since is compact, being homeomorphic to ). Call the union of these finitely many elements By the tube lemma, there is an open set of the form containing and contained in The collection of all for is an open cover of and hence has a finite subcover . Thus the finite collection covers . Using the fact that each is contained in and each is the finite union of elements of , one gets a finite subcollection of that covers .
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. The tube lemma follows from the generalized tube lemma by taking and It therefore suffices to prove the generalized tube lemma.