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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], with $Y$ [[compact|0TVJZYB6SH]], and give $X \\times Y$ the [[product topology|0GBG8B49AA]].\nLet $x_0 \\in X$, and let $N$ be an [[open set|02BH3MCBQB]] of $X \\times Y$ with $\\set{x_0} \\times Y \\subseteq N$.\nThen there is an open set $W \\subseteq X$ with $x_0 \\in W$ and $W \\times Y \\subseteq N$.\n\n## Remarks\n\nFor each $y \\in Y$, there is a basic open box $U_y \\times V_y \\subseteq N$ that contains $(x_0, y)$.\nThe sets $V_y$ cover the compact space $Y$, so finitely many $V_{y_1}, \\ldots, V_{y_n}$ cover it.\nThen $W = U_{y_1} \\cap \\cdots \\cap U_{y_n}$ is open, contains $x_0$, and satisfies $W \\times Y \\subseteq N$.\n\nThe lemma gives the proof that the [[product of two compact spaces is compact|09ZKE5XQ74]].\nIt also shows that the projection $X \\times Y \\to X$ is a [[closed map|00FSJTXPJM]] when $Y$ is compact.\n\nCompactness of $Y$ is necessary.\nLet $X = Y = \\R$, $x_0 = 0$, and $N = \\set{(x, y) : \\abs{xy} \u003c 1}$.\nThen $\\set{0} \\times \\R \\subseteq N$, but for any $\\varepsilon \u003e 0$ the point $(\\varepsilon / 2, 4 / \\varepsilon)$ lies in $(-\\varepsilon, \\varepsilon) \\times \\R$ and not in $N$.\n\n## Explanations\n\n### L1 human\n\nPicture $X \\times Y$ with $X$ horizontal and $Y$ vertical, and a vertical slice $\\set{x_0} \\times Y$ inside an open set $N$.\nThe lemma says that a whole vertical tube around the slice also fits inside $N$, as long as $Y$ is compact.\nWithout compactness, $N$ can get thinner and thinner as you go up, and no tube fits.\n\n### L2 human\n\nThe tube lemma is the key step in the proof that a finite product of compact spaces is compact.\nA more general form says that if $A \\subseteq X$ and $B \\subseteq Y$ are compact and $A \\times B \\subseteq N$ with $N$ open, then there are open $U \\supseteq A$ and $V \\supseteq B$ with $U \\times V \\subseteq N$.\nThis general form is often used to show that compact sets in Hausdorff spaces can be separated by open sets.\n",
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