# Thread: Topology and Open Set Question

1. ## Topology and Open Set Question

Define a topology on $\Re$(by listing the open sets within it) that contains the open set (0,2) and (1, 3) and that contains as few open sets as possible.

2. Originally Posted by r2dee6
Define a topology on $\Re$(by listing the open sets within it) that contains the open set (0,2) and (1, 3) and that contains as few open sets as possible.
You basically need to go back to the definition of a topology.

If $\mathcal{T}$ is a topology on the set $X$,
then $X$ and $\emptyset$ are in $\mathcal{T}$

And if there are A, B are open in $\mathcal{T}$, then $A \cap B$ is open in $\mathcal{T}$ also.

And if there are A, B, ... are open in $\mathcal{T}$, then $A \cup B$ is open in $\mathcal{T}$ also.

Can you take it from here?

3. I think the smallest such topology contains 8 sets- but there are several correct answers.

4. Originally Posted by HallsofIvy
I think the smallest such topology contains 8 sets- but there are several correct answers.
Interesting. I have only come up with 6.

5. It's also important to know: If each set $\mathcal{A}_1,\mathcal{A}_2,\cdots,\mathcal{A}_n$ is open, then
$\mathcal{M}_1:=\bigcup_{k=1}^{n}\mathcal{A}_k$
$\mathcal{M}_2:=\bigcap_{k=1}^{n}\mathcal{A}_k$
are open, if closed then closed. There's also a simple proof.

6. Originally Posted by chabmgph
Interesting. I have only come up with 6.
You are completely right. (I did say "I think"!) For some reason I was thinking that every we would need to add something like $(-\infty, 3)$ and $(0, \infty)$ to get all real numbers but we don't- they are already in R.