# metric space sets

• Dec 11th 2010, 03:37 AM
1234567
metric space sets
Hi,

Attachment 20060

I just can not find a set.
• Dec 11th 2010, 03:54 AM
FernandoRevilla
A little help:

(i) $\displaystyle A$ closed, bounded and compact is possible.

Choose $\displaystyle A=\{0\}$

(ii) $\displaystyle A$ closed, not bounded and compact is impossible because:

$\displaystyle A$ is compact iff $\displaystyle A$ is closed and bounded.

...

Fernando Revilla
• Dec 11th 2010, 03:56 AM
Plato
With sixty other postings, you should understand that this is not a homework service nor is it a tutorial service. So you need to show some of your own work on this problem or explain what you do not understand about the question.

Here is a start. $\displaystyle [0,1]$ is closed, bounded, and compact.
$\displaystyle (0,\infty)$ is not closed, not bounded, and not compact.
• Dec 11th 2010, 04:01 AM
1234567
Quote:

Originally Posted by Plato
With sixty other postings, you should understand that this is not a homework service nor is it a tutorial service. So you need to show some of your own work on this problem or explain what you do not understand about the question.

Here is a start. $\displaystyle [0,1]$ is closed, bounded, and compact.
$\displaystyle (0,\infty)$ is not closed, not bounded, and not compact.

I have considered some cases:

1 A union of closed and bounded set atr compact.

i.e [0,1] U [2,3]

2. (0,1) not closed so not compact

3. {1/n, n=1,2,3,...} U {0} compact bounded and closed

4. empty set compact bounded and closed

5. set R of all real numbers. Not compact
• Dec 11th 2010, 05:04 AM
1234567
• Dec 11th 2010, 05:27 AM
Plato
Quote:

Originally Posted by 1234567

Think of a truth-table
$\displaystyle \begin{array}{*{20}c} {closed} &\vline & {bounded} &\vline & {compact} \\\hline Y &\vline & Y &\vline & Y \\ Y &\vline & Y &\vline & N \\ Y &\vline & N &\vline & Y \\ Y &\vline & N &\vline & N \\ N &\vline & Y &\vline & Y \\ N &\vline & Y &\vline & N \\ N &\vline & N &\vline & Y \\ N &\vline & N &\vline & N \\ \end{array}$
I gave you the answers for the first line and the last line.
• Dec 11th 2010, 07:18 AM
1234567
Quote:

Originally Posted by Plato
Think of a truth-table
$\displaystyle \begin{array}{*{20}c} {closed} &\vline & {bounded} &\vline & {compact} \\\hline Y &\vline & Y &\vline & Y \\ Y &\vline & Y &\vline & N \\ Y &\vline & N &\vline & Y \\ Y &\vline & N &\vline & N \\ N &\vline & Y &\vline & Y \\ N &\vline & Y &\vline & N \\ N &\vline & N &\vline & Y \\ N &\vline & N &\vline & N \\ \end{array}$
I gave you the answers for the first line and the last line.

Thank you so much i can finish this question up