Re: [分析] compact sets

看板Math作者sato186 (銀色轟炸機)時間13年前 (2011/02/22 17:51)推噓2(2推 0噓 0→)

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※ 引述《r4553280 (Q睿)》之銘言： : Suppose that {Vα} α belong to A,is a collection of nonempty open sets in X : which satisfies Vα∩Vβ = Φ(空集合) for all α≠βin A.Prove that if X is : separable,then A is countable. : 這是這課本的習題 : 後面沒給提示 : 感覺跟Lindelof's theorem 有關係 : 可是想不到該怎麼證 : 可以請教一下解答已經該如何思考解題的方向嗎? : 先謝謝各位高手~ 果然是個上進的好學弟XD Proof 　　Let Ｃ:={ Ｖ } and S = ∪Ｃ. Clearly, Ｃ is an open cover α α in A for A. Since the space X is separable, by Lindelöf's theorem, there is a countable cover Ｋ of Ｃ so that S is contained in ∪Ｋ. Hence ∪Ｃ = ∪Ｋ. By the assumption of Ｃ, we can conclude that Ｃ = Ｋ is countable. 　　　　　　　　　　　　♪ -- 19487 XX 2/14 - □ (已被oralsex5566刪除) <> 請洽它它板

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r4553280

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