please:

I need your help in these 2 Qs:

the pictureAttachment 19565

Printable View

- Nov 3rd 2010, 03:03 AMrqeeb2 questions in TOPOLOGY
please:

I need your help in these 2 Qs:

the pictureAttachment 19565 - Nov 3rd 2010, 05:45 AMPlato
The first problem is proved in two steps.

First, If $\displaystyle B$ is a compact set and $\displaystyle p\notin B$ then there are two disjoint open sets such $\displaystyle B\subset G~\&~p\in H$.

Second, each point in $\displaystyle A$ is not in $\displaystyle B$.

If $\displaystyle a\in A$ use part one to get disjoint open sets $\displaystyle G_a~\&~H_a$.

The compactness of $\displaystyle A$ gives a finite collection of each.

The intersection of the $\displaystyle G_x's $ is open and contains $\displaystyle B$.

Problem #2. Use the co-finite topology to show it is false. - Nov 4th 2010, 02:10 AMrqeeb
thank u

but could u please clear the second question more - Nov 4th 2010, 06:07 AMDrexel28
He's suggesting that you take $\displaystyle \mathbb{R}$ with the cofinite topology. The important thing being that the resulting top. space isn't first countable.