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        "quote": "拓扑空间的典型模型是实数轴与欧几里得平面 $\\mathbb { R } ^ { 2 }$。在实数轴 $\\mathbb{R}$ 上，<u class=\"concept-term\" data-concept-identity=\"urn:uuid:cd84b279-ed33-43fd-be4d-3bf0bfb28da5\" data-concept-route=\"/concept-identities#concept-cd84b279-ed33-43fd-be4d-3bf0bfb28da5\" id=\"first-8686d43e8087afd9ddfc-zh\">开区间(open interval)</u> 是任意集合 $( a , b ) = \\{ x \\in \\mathbb { R } | a < x < b \\}$。若存在 $\\epsilon > 0$ 使得开区间 $( x - \\epsilon , x + \\epsilon )$ 是 $U$ 的子集，则称集合 $U \\subseteq \\mathbb { R }$ 为 $x \\in \\mathbb { R }$ 的 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:ed082212-4450-43ff-be65-96e35fa3d058\" data-concept-route=\"/concept-identities#concept-ed082212-4450-43ff-be65-96e35fa3d058\" id=\"first-45266bf66f5e877e61c9-zh\">邻域(neighbourhood)</u>。我们说实数序列 $\\{ x _ { n } \\}$ <u class=\"concept-term\" data-concept-identity=\"urn:uuid:4630b373-c711-4f78-89ac-9d391ebb15dc\" data-concept-route=\"/concept-identities#concept-4630b373-c711-4f78-89ac-9d391ebb15dc\" id=\"first-db5608de19c5f800f7f2-zh\">收敛(convergence)</u>到 $x \\in \\mathbb { R }$，记作 $x _ { n } \\to x$，如果对每个 $\\epsilon > 0$ 都存在整数 $N > 0$，使得对所有 $n > N ;$ 有 $| x -x _ { n } | < \\epsilon$；也就是说，当 $n$ 充分大时，序列 $x _ { n }$ 进入并停留在 $x$ 的每个邻域 $U$ 中。此时称点 $x$ 为序列 $\\{ x _ { n } \\}$ 的 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:ea27b1d9-445a-474d-a590-170a34051242\" data-concept-route=\"/concept-identities#concept-ea27b1d9-445a-474d-a590-170a34051242\" id=\"first-47b0f6a99aa530cc1f8c-zh\">极限(limit)</u>。",
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