Topology and Open Set Question

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January 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.
January 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?
January 12th 2009, 05:51 AM
HallsofIvy

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