f*****g
发帖数: 9098 | 1 如何证明
The image of a borel set under a continuous 1-1 map is again a borel set (in
R^n setting)? |
j******w
发帖数: 690 | 2 This is not that trivial.
Here is a recursion theoretic proof:
Suppose A is a hyperarithmetic set and f is a recursive function which is 1-
1 on A, then the image B of A
under f is a \Sigma^1_1 set.
To see that B is \Pi^1_1, just notice that y \in B iff there exists a real x
hyperarithmetic in y so that f(x)=y
(this is because \{x\} is a \Sigma^1_1(y) set).
So B is a hyperarithmetic set.
Relativizing the proof, it is easy to show the result.
A pure set theoretic proof can be found in Kechris' bo
【在 f*****g 的大作中提到】
: 如何证明
: The image of a borel set under a continuous 1-1 map is again a borel set (in
: R^n setting)?
|
f*****g
发帖数: 9098 | 3 原来还要引入其他概念，俺还没到那个水平。。。
像continuous functions, borel sets都是很初级的概念，但为什么可以用最基本的实
分析的语言来描述的命题，证明却要借助更高级的概念？
还有，下面这个较弱的命题会不会有简单一点的证明？
The image of an open set under a continuous 1-1 map is a Borel set?
1-
x
【在 j******w 的大作中提到】
: This is not that trivial.
: Here is a recursion theoretic proof:
: Suppose A is a hyperarithmetic set and f is a recursive function which is 1-
: 1 on A, then the image B of A
: under f is a \Sigma^1_1 set.
: To see that B is \Pi^1_1, just notice that y \in B iff there exists a real x
: hyperarithmetic in y so that f(x)=y
: (this is because \{x\} is a \Sigma^1_1(y) set).
: So B is a hyperarithmetic set.
: Relativizing the proof, it is easy to show the result.
|
H****h
发帖数: 1037 | 4 开集可以写成可数个紧集的并集。
【在 f*****g 的大作中提到】
: 原来还要引入其他概念，俺还没到那个水平。。。
: 像continuous functions, borel sets都是很初级的概念，但为什么可以用最基本的实
: 分析的语言来描述的命题，证明却要借助更高级的概念？
: 还有，下面这个较弱的命题会不会有简单一点的证明？
: The image of an open set under a continuous 1-1 map is a Borel set?
:
: 1-
: x
|
r*******y
发帖数: 1081 | 5 a continuous function pull back an open set to an open set.
in
【在 f*****g 的大作中提到】
: 如何证明
: The image of a borel set under a continuous 1-1 map is again a borel set (in
: R^n setting)?
|
j******w
发帖数: 690 | 6 come on.
Lots of Number theory problems (GH, Fermat theorem...) look stupid but they
require really much, right?
"The image of an open set under a continuous 1-1 map is a Borel set" Even
this is not trivial if you consider Baire Space (For R^n, the nontrivial case shows up at G_{\delta} level).
Kechris' proof require much fewer but it's weaker.
Actually the previous proof shows more:
If A is a Borel set and f is a borel function with countable to 1, then the
image of A under f is a Borel set.
I
【在 f*****g 的大作中提到】
: 原来还要引入其他概念，俺还没到那个水平。。。
: 像continuous functions, borel sets都是很初级的概念，但为什么可以用最基本的实
: 分析的语言来描述的命题，证明却要借助更高级的概念？
: 还有，下面这个较弱的命题会不会有简单一点的证明？
: The image of an open set under a continuous 1-1 map is a Borel set?
:
: 1-
: x
|
f*****g
发帖数: 9098 | 7 赞，我也想到了，很好的利用了R^n的假设。
【在 H****h 的大作中提到】
: 开集可以写成可数个紧集的并集。
|
f*****g
发帖数: 9098 | 8 does it push forward an open set to an open set?
【在 r*******y 的大作中提到】
: a continuous function pull back an open set to an open set.
:
: in
|
H****h
发帖数: 1037 | 9 如果是可微函数，那么可以推出加科比处处非零，所以把开集映为开集。
【在 f*****g 的大作中提到】
: does it push forward an open set to an open set?
|
f*****g
发帖数: 9098 | 10 open mapping theorem？ 少条件吧
【在 H****h 的大作中提到】
: 如果是可微函数，那么可以推出加科比处处非零，所以把开集映为开集。
|
r*******y
发帖数: 1081 | 11 no.
【在 f*****g 的大作中提到】
: does it push forward an open set to an open set?
|
o****e
发帖数: 92 | 12 你这话不知做数论的听了有什么感想，lolz
【在 f*****g 的大作中提到】
: 原来还要引入其他概念，俺还没到那个水平。。。
: 像continuous functions, borel sets都是很初级的概念，但为什么可以用最基本的实
: 分析的语言来描述的命题，证明却要借助更高级的概念？
: 还有，下面这个较弱的命题会不会有简单一点的证明？
: The image of an open set under a continuous 1-1 map is a Borel set?
:
: 1-
: x
|
