Results 1 to 4 of 4

Thread: Help understanding this theorem/proof

  1. #1
    Member
    Joined
    Mar 2009
    Posts
    168

    Help understanding this theorem/proof

    Here's the theorem that I have to prove:

    A set $\displaystyle U \subset R$ is called an open set if $\displaystyle \forall x \in U \ \exists \epsilon >0$ such that $\displaystyle (x-\epsilon, x+\epsilon) \subset U$. Prove that the following two statements are equivalent:
    (a) $\displaystyle f \rightarrow R$ is continuous
    (b) For any open set $\displaystyle U \subset R$, there exists an open set $\displaystyle W \subset R$ such that $\displaystyle f^{-1}(U)=D \cap W$. Here, $\displaystyle f^{-1}(U)=\{x \in D|f(x) \in U\}$ denotes the preimage of U under f.

    Can anyone help me with this? I don't even understand the "idea" of it, much less how to prove it.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Mar 2009
    Posts
    168
    Anyone able to help with this? Does this theorem have a name, by chance?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Focus's Avatar
    Joined
    Aug 2009
    Posts
    228
    Quote Originally Posted by paupsers View Post
    Here's the theorem that I have to prove:

    A set $\displaystyle U \subset R$ is called an open set if $\displaystyle \forall x \in U \ \exists \epsilon >0$ such that $\displaystyle (x-\epsilon, x+\epsilon) \subset U$. Prove that the following two statements are equivalent:
    (a) $\displaystyle f \rightarrow R$ is continuous
    (b) For any open set $\displaystyle U \subset R$, there exists an open set $\displaystyle W \subset R$ such that $\displaystyle f^{-1}(U)=D \cap W$. Here, $\displaystyle f^{-1}(U)=\{x \in D|f(x) \in U\}$ denotes the preimage of U under f.

    Can anyone help me with this? I don't even understand the "idea" of it, much less how to prove it.
    Which statement of continuity do you know? b) is a pretty standard definition of continuity from a subspace topology. I am guessing that D is a subset of R with the subspace topology.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Dec 2009
    Posts
    3
    bump
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Understanding the Cauchy Integral Theorem
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Nov 12th 2011, 02:05 PM
  2. Help understanding what this theorem means
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Apr 9th 2011, 01:48 AM
  3. Need help understanding proof of theorem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Mar 8th 2010, 09:41 AM
  4. Problem understanding the theorem
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: Jan 5th 2010, 08:30 PM
  5. Need help understanding the Chinese Remainder Theorem
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: Nov 18th 2009, 02:46 PM

Search Tags


/mathhelpforum @mathhelpforum