Define a topology on $\displaystyle \Re$(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.

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- Jan 11th 2009, 02:33 PMr2dee6Topology and Open Set Question
Define a topology on $\displaystyle \Re$(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 PMchabmgph
You basically need to go back to the definition of a topology.

If $\displaystyle \mathcal{T}$ is a topology on the set $\displaystyle X$,

then $\displaystyle X$ and $\displaystyle \emptyset$ are in $\displaystyle \mathcal{T}$

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

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

Can you take it from here? - Jan 12th 2009, 05:51 AMHallsofIvy
I

**think**the smallest such topology contains 8 sets- but there are several correct answers. - Jan 12th 2009, 07:59 AMchabmgph
- Jan 12th 2009, 08:08 AMlukaszh
It's also important to know: If each set $\displaystyle \mathcal{A}_1,\mathcal{A}_2,\cdots,\mathcal{A}_n$ is open, then

$\displaystyle \mathcal{M}_1:=\bigcup_{k=1}^{n}\mathcal{A}_k$

$\displaystyle \mathcal{M}_2:=\bigcap_{k=1}^{n}\mathcal{A}_k$

are open, if closed then closed. There's also a simple proof. - Jan 12th 2009, 10:00 AMHallsofIvy