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        "quote": "集合 $X \\times Y$ 上的 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:8bed2d93-396b-4829-9d37-d34ecc7e93b7\" data-concept-route=\"/concept-identities#concept-8bed2d93-396b-4829-9d37-d34ecc7e93b7\" id=\"first-dc3eb7b09be2b856c40b-zh\">积拓扑(product topology)</u> 定义为由这两个映射诱导的拓扑。带有积拓扑的空间 $X \\times Y$ 称为 $X$ 与 $Y$ 的 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:73c98f07-9d4e-4cc9-bae7-adbe213355fe\" data-concept-route=\"/concept-identities#concept-73c98f07-9d4e-4cc9-bae7-adbe213355fe\" id=\"first-f447e5ade7d3cab2b68d-zh\">拓扑积(topological product)</u>。它是使得各投影映射连续的最粗拓扑。$X$ 中开集 $U$ 在 $\\operatorname{pr} _ { 1 }$ 下的原像是“竖直带”$U \\times Y$，而 $Y$ 中开集 $V$ 在 $\\operatorname{pr} _ { 2 }$ 下的原像是“水平带”$X \\times V$。任意一对这样的带的交集是形如 $U \\times V$ 的集合，其中 $U$ 与 $V$ 分别是来自 $X$ 与 $Y$ 的开集（见图 10.4）。由于由竖直带与水平带生成的拓扑由所有这些交集的任意并组成，因此在积拓扑中，子集 $A \\subset X \\times Y$ 是开集，当且仅当对每一点 $( x , y ) \\in A$ 都存在开集 $U \\in { \\mathcal { O } } _ { X }$ 与 $V \\in { \\mathcal { O } } _ { Y }$，使得 $( x , y ) \\in U \\times V \\subseteq A$",
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