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.
compact, and consider the product space
is an open set containing a slice in
Using the concept of closed maps, this can be rephrased concisely as follows: if
a compact space, then the projection map
Generalized Tube Lemma 1 — Let
is an open set containing
Generalized Tube Lemma 2 — Let
is an open set containing
in the product topology, that is the Euclidean plane, and the open set
The open set
but contains no tube, so in this case the tube lemma fails.
contradicting the fact that
This shows that the compactness assumption is essential.
The tube lemma can be used to prove that if
are compact spaces, then
Call the union of these finitely many elements
By the tube lemma, there is an open set of the form
Thus the finite collection
is the finite union of elements of
By part 2 and induction, one can show that the finite product of compact spaces is compact.
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
It therefore suffices to prove the generalized tube lemma.
By the definition of the product topology, for each
there are open sets
is an open cover of the compact set
so this cover has a finite subcover; namely, there is a finite set
We now essentially repeat the argument to drop the dependence on
are open, which completes the proof.