Let $A = \{0\}\cup\{1, 1/2, ..., 1/n, ...\}$
I would like show that A directly satisfies the definition of compactness which is "every open cover has a finite sub-cover"
How could one generally select open cover of $A$ to start the proof?
Asked
Active
Viewed 61 times
0
-
If $U$ is an open set containing $0$. How many points are in $A-U$? – jJjjJ Jun 02 '17 at 07:21
-
Take any open cover $\mathcal{B}={B_1,B_2,...,}$ of $A$. Any one of $B_i$ covers $0$ .What can you say about this $B_i$? – Chinnapparaj R Jun 02 '17 at 07:22
-
@iJjjJ Doesn't it depend on U? I mean differ from case to case. – delog Jun 02 '17 at 07:23
-
Yes but the fact is: are they finite or not? – jJjjJ Jun 02 '17 at 07:24
-
@iJjjJ infinite – delog Jun 02 '17 at 07:26
-
Read again my first comment and think carefully – jJjjJ Jun 02 '17 at 07:27
-
@jJjjJ sorry. it's finite since 1/n -> 0 as n-> $\infty$ – delog Jun 02 '17 at 07:30