text: # 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