2 questions in TOPOLOGY

**rqeeb** · November 3rd 2010, 03:03 AM

please:
I need your help in these 2 Qs:
the picture

**Plato** · November 3rd 2010, 05:45 AM

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

Second, each point in $A$ is not in $B$ .
If $a\in A$ use part one to get disjoint open sets $G_a~\&~H_a$ .
The compactness of $A$ gives a finite collection of each.
The intersection of the $G_x's$ is open and contains $B$ .

Problem #2. Use the co-finite topology to show it is false.

**rqeeb** · November 4th 2010, 02:10 AM

thank u
but could u please clear the second question more

**Drexel28** · November 4th 2010, 06:07 AM

Originally Posted by rqeeb

thank u
but could u please clear the second question more

He's suggesting that you take $\mathbb{R}$ with the cofinite topology. The important thing being that the resulting top. space isn't first countable.

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