# Tube lemma

## Statement

Let $X$ be a compact space and $A$ any topological space. Consider $X \times A$ endowed with the product topology. Suppose $a \in A$ and $U$ is an open subset of $X \times A$ containing the entire slice $X \times \{ a \}$. Then, we can find an open subset $V$ of $A$ such that:

$a \in V$, and $X \times V \subseteq U$

In other words, any open subset containing a slice contains an open cylinder that contains the slice.

## Proof

Given: A compact space $X$, a topological space $A$. $a \in A$, and $U$ is an open subset of $X \times A$ containing the slice $X \times \{ a \}$.

To prove: There exists an open subset $V$ of $A$ such that $a \in V$ $X \times V$ is contained in $U$.

Proof:

1. A collection of open subsets inside $U$ whose union contains $X \times \{ a \}$: For each $x \in X$, we have $(x,a) \in U$, so by the definition of openness in the product topology, there exists a basis open subset $M_x \times N_x \subseteq U$ containing $(x,a)$. In particular, we get a collection $M_x \times N_x, x \in X$ of open subsets contained in $U$, whose union contains $X \times \{ a \}$.
2. This collection yields a point-indexed open cover for $X$: Note that since $M_x \times N_x$ is a basis open set containing $(x,a)$, $M_x$ is an open subset of $X$ containing $X$, so the $M_x, x \in X$, form an open cover of $X$.
3. (Given data used: $X$ is compact): This cover has a finite subcover: Indeed, since $X$ is compact, we can choose a finite collection of points $\{ x_1, x_2, \dots, x_n \} \subseteq X$ such that $X$ is the union of the $M_{x_i}$s.
4. If $V$ is the intersection of the corresponding $N_{x_i}$s, then $V$ is open in $A$, $a \in V$, and $X \times V \subseteq U$: First, $V$ is open since it is an intersection of finitely many open subsets of $Y$. Second, each $N_{x_i}$ contains $a$, so $a \in V$. Third, if $(x,v) \in X \times V$, then there exists $x_i$ such that $x \in M_{x_i}$. By definition, $v \in N_{x_i}$, so $(x,v) \in M_{x_i} \times N_{x_i} \subseteq V$. Thus, $(x,v) \in U$.

## References

### Textbook references

• Topology (2nd edition) by James R. Munkres, More info, Page 168, Lemma 26.8, Chapter 3, Section 26 (the proof is given before the theorem, as Step 1 of the proof of Theorem 26.7 on page 167)