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        "quote": "> [!example] <a class=\"object-permalink\" href=\"#example-1-14\">例 1.14</a> · 拓扑\n>\n> $S$ 上的<u class=\"concept-term\" data-concept-identity=\"urn:uuid:1ebf8613-2556-44a4-854a-86128769ba52\" data-concept-route=\"/concept-identities#concept-1ebf8613-2556-44a4-854a-86128769ba52\" id=\"first-2564e08cd4afded1d225-zh\">拓扑(topology)</u>是 $\\mathcal O\\subseteq2^S$，其成员称为<u class=\"concept-term\" data-concept-identity=\"urn:uuid:c3ac7e5b-190b-46ab-86f5-5288c831eecc\" data-concept-route=\"/concept-identities#concept-c3ac7e5b-190b-46ab-86f5-5288c831eecc\" id=\"first-6c8a28b7ca0c9a5df668-zh\">开集(open set)</u>，满足：\n>\n> 1. $\\varnothing,S\\in\\mathcal O$。\n> 2. 若 $U,V\\in\\mathcal O$，则 $U\\cap V\\in\\mathcal O$。\n> 3. 若 $\\mathcal U\\subseteq\\mathcal O$，则 $\\bigcup\\mathcal U\\in\\mathcal O$。\n>\n> 第二条推出有限交仍开；第三条允许任意并，包括无限并。它显然属于对子集族作规定的几何结构。原书第 10 章系统讨论拓扑。",
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        "quote": "在 $\\mathbb{R}$ 或 $\\mathbb { R } ^ { 2 }$ 中的 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:c3ac7e5b-190b-46ab-86f5-5288c831eecc\" data-concept-route=\"/concept-identities#concept-c3ac7e5b-190b-46ab-86f5-5288c831eecc\">开集(open set)</u> $U$ 是一个集合，它是其中每一点的邻域。直观地说，$U$ 在 $\\mathbb{R}$ 中是开的（相应地，$\\mathbb { R } ^ { 2 } )$，如果 $U$ 中的每一点都可以被“加厚”为 $U$ 内的一个开区间（相应地，开球）（见图 10.1）。例如，单位球 $B _ { 1 } ( { \\bf O } ) = \\{ { \\bf y } | | { \\bf y } | ^ { 2 } < 1 \\}$ 是一个开集，因为对于每一点 $\\mathbf { x } \\in B _ { 1 } ( \\mathbf { O } )$，开球 $B _ { \\epsilon } ( { \\bf x } ) \\subseteq B _ { 1 } ( { \\bf O } )$，其中 $\\epsilon =1 - | \\mathbf { x } | > 0$",
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