Lemma 4.16.5 (Tube Lemma). Let $X$ be a topological space and $Y$ be a compact space, then
For any $x \in X$ and $U \in \cn_{X \times Y}^{o}(\bracs{x}\times Y)$, there exists $V \in \cn_{X}(x)$ such that $V \times Y \subset U$.
For any $A \subset X$ and $U \in \cn_{X \times Y}^{o}(A \times Y)$, there exists $V \in \cn_{X}(A)$ such that $V \times Y \subset U$.
Proof. (1): For each $y \in Y$, let $U_{y} \in \cn_{X}(x)$ and $V_{y} \in \cn_{Y}(y)$ such that $(x, y) \in U_{y} \times V_{y} \subset U$, then $\bracs{V_y|y \in Y}$ is an open cover of $Y$. By compactness, there exists $\seqf{y_j}\subset Y$ such that $\bigcup_{j = 1}^{n}V_{y_j}= Y$. Let $V = \bigcap_{j = 1}^{n} U_{y_j}$, then $V \in \cn_{X}(x)$ and $V \times Y \subset U$.
(2): For each $x \in A$, there exists $U_{x} \in \cn_{X}(x)$ such that $U_{x} \times Y \subset U$. Let $V = \bigcup_{x \in A}U_{x}$, then $V \times Y \subset U$.$\square$