tube lemma

Tube lemma - Let X and Y be topological spacesMathworldPlanetmath such that Y is compactPlanetmathPlanetmath. If N is an open set of X×Y containing a ”slice” x0×Y, then N contains some ”tube” W×Y, where W is a neighborhoodMathworldPlanetmathPlanetmath of x0 in X.

Proof : N is a union of basis elements U×V, with U and V open sets in X and Y respect. Since x0×Y is compact (it is homeomorphic to Y), only a finite number U1×V1,,Un×Vn of such basis elements cover x0×Y.

We may assume that each of the basis elements Ui×Vi actually intersects x0×Y, since otherwise we could discard it from the finite collectionMathworldPlanetmath and still have a covering of x0×Y.

Define W:=U1Un. The set W is open and contains x0 because each Ui×Vi intersects x0×Y by the previous remark.

We now claim that W×YN. Let (x,y) be a point in W×Y. The point (x0,y) is in some Ui×Vi and so yVi. We also know that xW=U1UnUi.

Therefore (x,y)Ui×ViN as desired.


  • 1 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
Title tube lemma
Canonical name TubeLemma
Date of creation 2013-03-22 17:25:39
Last modified on 2013-03-22 17:25:39
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Theorem
Classification msc 54D30