# Topology and Open Set Question

• Jan 11th 2009, 02:33 PM
r2dee6
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.
• Jan 11th 2009, 03:11 PM
chabmgph
Quote:

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?
• Jan 12th 2009, 05:51 AM
HallsofIvy
I think the smallest such topology contains 8 sets- but there are several correct answers.
• Jan 12th 2009, 07:59 AM
chabmgph
Quote:

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. (Itwasntme)
• Jan 12th 2009, 08:08 AM
lukaszh
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.
• Jan 12th 2009, 10:00 AM
HallsofIvy
Quote:

Originally Posted by chabmgph
Interesting. I have only come up with 6. (Itwasntme)

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.