# Tube lemma

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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 $X \times V \subseteq A$