Continuous & open map

[rcharity]

proof or disproof

f:X → Y

f^-1 : continuous iff f: open map

 

[Deveno]

I will assume it is given that $f$ is invertible, since otherwise saying $f^{-1}$ is continuous is meaningless.

Now $f^{-1}$ continuous means that for any open set $U \subseteq X$,

$(f^{-1})^{-1}(U)$ is an open set in $Y$

(here the first inverse sign means "inverse" and the outer one means "pre-image", however: since $f$ is invertible these are the same).

thus $f(U) = (f^{-1})^{-1}(U)$ is open for any open set $U$, that is: $f$ is open.

Can you continue?

 

[rcharity]

your right
Thank you

내가 영어를 조금 더 잘했다면 많은 이야기를 했을텐데
영어를 못해서 읽고 이해할수는 있는데 표현을 못해

나 역시 계속 생각해보니 f가 가역이라는 가정이 들어가면 참이라고 생각해~
아무튼 고맙다~

 

[MarkFL]

your right
Thank you

내가 영어를 조금 더 잘했다면 많은 이야기를 했을텐데
영어를 못해서 읽고 이해할수는 있는데 표현을 못해

나 역시 계속 생각해보니 f가 가역이라는 가정이 들어가면 참이라고 생각해~
아무튼 고맙다~

Hello rcharity,

We ask that posts be made in English here, as this is language we have chosen to bind together our community. So please, in the future use English to convey your thoughts so that everyone here knows what you are saying. Thank you!

 

[rcharity]

Hello rcharity,

We ask that posts be made in English here, as this is language we have chosen to bind together our community. So please, in the future use English to convey your thoughts so that everyone here knows what you are saying. Thank you!

I know posts be mada in English.
But, My English is beginner.
So I can read English but, I have difficult to my think presentation.
other language mean is your right and Thank you.
I hope say detaily this feeling use my native language.
you want not, I will not say native language.

Maybe, do you understand my native language?

 

[MarkFL]

I commend you for learning another language and can appreciate the difficulty that I myself would have trying to communicate about anything, especially mathematics, in any language other than English.

We certainly will not fault you for any grammatical errors you may make, so please do not feel like your English must meet a certain level of proficiency or that we will be critical. I am certain we will be able to read and understand your posts.