a***n
发帖数: 202 | 1 I am confused with compact and locally compact.
As the image of compact set is compact under continuous map.
Does the image of locally compact space is locally compact under a continuous
map f?
what if f is both continuous and open?
This is a question from the textbook, not homework. But I have thinked about
it for several days.
can anybody clearify this for me? thanks! |
a***n
发帖数: 202 | 2 有谁能给点提示么？想了好几天了。
continuous
【在 a***n 的大作中提到】
: I am confused with compact and locally compact.
: As the image of compact set is compact under continuous map.
: Does the image of locally compact space is locally compact under a continuous
: map f?
: what if f is both continuous and open?
: This is a question from the textbook, not homework. But I have thinked about
: it for several days.
: can anybody clearify this for me? thanks!
|
B****n
发帖数: 11290 | 3 if f is continuous and open mapping, the the image must also
be locally compact. it's just from the definition of local compact.
【在 a***n 的大作中提到】
: 有谁能给点提示么？想了好几天了。
:
: continuous
|
B****n
发帖数: 11290 | 4 If f is continuous but not open mapping, then the image is not necessarily
locally compact. ex: f:[-inf,inf]*[-inf,inf]->L2[-pi,pi] with topology induced
by the matric sqrt(integral(f*g)^2)
Define f(x,y)=y*sin(x*t) then f is continuous but {y*sin(x*t)} is not locally
compact because every closed ball contains a set of infinite orthogonal
functions; hence not every sequence can find convergent subsequence.
【在 B****n 的大作中提到】
: if f is continuous and open mapping, the the image must also
: be locally compact. it's just from the definition of local compact.
|
c****n
发帖数: 2031 | 5 shall we also assume that f(0)=0
【在 B****n 的大作中提到】
: if f is continuous and open mapping, the the image must also
: be locally compact. it's just from the definition of local compact.
|
a***n
发帖数: 202 | 6 Hi, Thank you very much!
I just think about another counter-example: the cantor function on [0,1].
Thank you!
induced
locally
【在 B****n 的大作中提到】
: If f is continuous but not open mapping, then the image is not necessarily
: locally compact. ex: f:[-inf,inf]*[-inf,inf]->L2[-pi,pi] with topology induced
: by the matric sqrt(integral(f*g)^2)
: Define f(x,y)=y*sin(x*t) then f is continuous but {y*sin(x*t)} is not locally
: compact because every closed ball contains a set of infinite orthogonal
: functions; hence not every sequence can find convergent subsequence.
|
B****n
发帖数: 11290 | 7 Is the image of cantor function not locally compact?
May I ask how do you define the cantor function and what is its image?
thanks
【在 a***n 的大作中提到】
: Hi, Thank you very much!
: I just think about another counter-example: the cantor function on [0,1].
: Thank you!
:
: induced
: locally
|
