# tube lemma

Let $X$ and $Y$ be topological spaces such that $Y$ is compact. If $N$ is an open set of $X\times Y$ containing a ”slice” $x_{0}\times Y$, then $N$ contains some ”tube” $W\times Y$, where $W$ is a neighborhood of $x_{0}$ in $X$.

Proof : $N$ is a union of basis elements $U\times V$, with $U$ and $V$ open sets in $X$ and $Y$ respect. Since $x_{0}\times Y$ is compact (it is homeomorphic to $Y$), only a finite number $U_{1}\times V_{1},\dots,U_{n}\times V_{n}$ of such basis elements cover $x_{0}\times Y$.

We may assume that each of the basis elements $U_{i}\times V_{i}$ actually intersects $x_{0}\times Y$, since otherwise we could discard it from the finite collection and still have a covering of $x_{0}\times Y$.

Define $W:=U_{1}\cap\dots\cap U_{n}$. The set $W$ is open and contains $x_{0}$ because each $U_{i}\times V_{i}$ intersects $x_{0}\times Y$ by the previous remark.

We now claim that $W\times Y\subseteq N$. Let $(x,y)$ be a point in $W\times Y$. The point $(x_{0},y)$ is in some $U_{i}\times V_{i}$ and so $y\in V_{i}$. We also know that $x\in W=U_{1}\cap\dots\cap U_{n}\subseteq U_{i}$.

Therefore $(x,y)\in U_{i}\times V_{i}\subseteq N$ as desired. $\square$

## References

• 1 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
Title tube lemma TubeLemma 2013-03-22 17:25:39 2013-03-22 17:25:39 asteroid (17536) asteroid (17536) 7 asteroid (17536) Theorem msc 54D30