Thread: Help understanding this theorem/proof

1. Help understanding this theorem/proof

Here's the theorem that I have to prove:

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

2. Anyone able to help with this? Does this theorem have a name, by chance?

3. Originally Posted by paupsers
Here's the theorem that I have to prove:

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

4. bump