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        "quote": "这种诱导拓扑的一个常见实例出现在拓扑空间 $X$ 上定义了一个等价关系 $E$ 的时候。设 $[ x ] = \\{ y \\mid y E x \\}$ 为包含点 $x \\in X$ 的等价类。在商空间 $X / E = \\{ [ x ] | x \\in X \\}$ 中，定义从 $X$ 到 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:f5be2df2-a716-470d-93b8-2043adf53100\" data-concept-route=\"/concept-identities#concept-f5be2df2-a716-470d-93b8-2043adf53100\" id=\"first-b36dbe6e996de25a22e9-zh\">粘合拓扑(identification topology)</u> 的拓扑为由自然映射 $i_E: X \\to X/E$ 诱导的拓扑，该映射将每个点 $x \\in X$ 对应于它所属于的等价类 $i _ { E } ( x ) = [ x ]$。在此拓扑中，子集 $A$ 是开集当且仅当其逆像 $i _ { E } ^ { - 1 } ( A ) = \\{ x \\in X \\mid [ x ] \\in\\begin{array} { r } { A \\} = \\bigcup _ { [ x ] \\in A } [ x ] } \\end{array}$ 是 $X$ 的开子集。也就是说，等价类的子集 $A = \\{ [ x ] \\}$ 在 $X / E$ 上的粘合拓扑中是开集，当且仅当属于 $A$ 的集合 $[x]$ 的并集是 $X$ 的开子集。",
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