# 2 questions in TOPOLOGY

• Nov 3rd 2010, 03:03 AM
rqeeb
2 questions in TOPOLOGY
I need your help in these 2 Qs:
the pictureAttachment 19565
• Nov 3rd 2010, 05:45 AM
Plato
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 AM
rqeeb
thank u
but could u please clear the second question more
• Nov 4th 2010, 06:07 AM
Drexel28
Quote:

Originally Posted by rqeeb
thank u
but could u please clear the second question more

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.