Let $ L$ be closed, then $ \displaystyle \bigcap_{n=1}^\infty \left(\bigcup_{x\in L}B\left(x,\frac{1}{n}\right)\right)=L$ (i.e. $ L$ is a $ G_\delta$ set).
李教授話我個 proof 睇落去冇問題，咁即係佢 expect ge 證法唔係咁喇 ... sosad。
thx, in class we are discussing stuff in metric space. The tool I used in the proof is the proved fact $latex d(x,A)=0\implies x\in \overline{A}$.
But we also know that the intersection and union operator can be interchanged, and as $latex \displaystyle \bigcap_{n=1}^\infty B\left(x,\frac{1}{n}\right)=\{x\}$, taking the union on both sides will show all sets in metric space is a $latex G_\delta$ set, is it correct?...
This is wrong. In fact, arbitrary intersection and union CANNOT be interchanged. (eg, see wikipedia: http://en.wikipedia.org/wiki/Union_(set_theory), last statement)
I can quickly explain the intuition behind this discrepancy as follows. Recall the limsup and liminf of sequences $latex x_n$ can be written as $latex limsup x_n = \inf_k sup_{n \ge k} x_n$ and $latex liminf x_n = \sup_k inf_{n \ge k} x_n$.
In general, inf feels like intersection, sup feels like union. First intersect, then union feels like liminf. First union, then intersect feels like limsup. You know that in general these two can't be the same. (e.g., see http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior#Sequences_of_sets)
Therefore your argument is wrong. In general if your L is not closed, then LHS of your equality = closure of L.
Another way to spot the problem is that you cannot expect all sets to be G delta. For example, using Baire category theorem (which you will learn in either 370 or 371), one has the following result (see Royden, Real Analysis P. 161): If $latex E$ is a subset of a complete metric space such that both $latex E$ and its complement is dense, then at most one of them is $latex G_{\delta}$. This tells you, for example, that the rational numbers in $latex [0,1]$ is not a $latex G_{\delta}$ set. (You can try to think about the reason yourself)
``For example, using Baire category theorem (which you will learn in either 370 or 371)" 其實我呢個 summer reg 左 370 ... =_=，你唔講我都醒唔起 royden 有 abstract space 呢課 ...。
提到 liminf 同 limsup，下個 sem 我一定會 sit 返齊 301, 特別 limsup, liminf, uniform convergence 呢幾 part，可能我吸收得太慢，203 唔知學左 d 乜黎 = =。
ur statement is right for a metric space.
ReplyDeletethx, in class we are discussing stuff in metric space. The tool I used in the proof is the proved fact $latex d(x,A)=0\implies x\in \overline{A}$.
ReplyDeleteBut we also know that the intersection and union operator can be interchanged, and as $latex \displaystyle \bigcap_{n=1}^\infty B\left(x,\frac{1}{n}\right)=\{x\}$, taking the union on both sides will show all sets in metric space is a $latex G_\delta$ set, is it correct?...
This is wrong. In fact, arbitrary intersection and union CANNOT be interchanged. (eg, see wikipedia: http://en.wikipedia.org/wiki/Union_(set_theory), last statement)
ReplyDeleteI can quickly explain the intuition behind this discrepancy as follows. Recall the limsup and liminf of sequences $latex x_n$ can be written as
$latex limsup x_n = \inf_k sup_{n \ge k} x_n$ and $latex liminf x_n = \sup_k inf_{n \ge k} x_n$.
In general, inf feels like intersection, sup feels like union.
First intersect, then union feels like liminf.
First union, then intersect feels like limsup.
You know that in general these two can't be the same. (e.g., see http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior#Sequences_of_sets)
Therefore your argument is wrong. In general if your L is not closed, then LHS of your equality = closure of L.
Another way to spot the problem is that you cannot expect all sets to be G delta. For example, using Baire category theorem (which you will learn in either 370 or 371), one has the following result (see Royden, Real Analysis P. 161):
If $latex E$ is a subset of a complete metric space such that both $latex E$ and its complement is dense, then at most one of them is $latex G_{\delta}$. This tells you, for example, that the rational numbers in $latex [0,1]$ is not a $latex G_{\delta}$ set. (You can try to think about the reason yourself)
死... 原來係我癲左，我果時都有 o係 wiki 睇過但唔知乜事當左係等號...。
ReplyDelete``For example, using Baire category theorem (which you will learn in either 370 or 371)"
其實我呢個 summer reg 左 370 ... =_=，你唔講我都醒唔起 royden 有 abstract space 呢課 ...。
提到 liminf 同 limsup，下個 sem 我一定會 sit 返齊 301, 特別 limsup, liminf, uniform convergence 呢幾 part，可能我吸收得太慢，203 唔知學左 d 乜黎 = =。
Royden 有講measure space，但baire category theorem唔關事，呢個係complete metric space (or locally compact Hausdorff space)既結果，不過royden part 1(定part 2，唔記得)都有。
ReplyDeleteliminf/limsup, 建議你溫203，比較conceptual。301計算比較多，但唔conceptual，對你以後學黎講203果度會好d