Tube lemma

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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.