Tube lemma

This fact is related to: compactness

This article is about the statement of a simple but indispensable lemma in topology

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 $X\times V\subseteq A$

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

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)