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
explanation: The document is a discussion thread from a math forum, where a user is asking for help with a problem related to metric space sets, specifically about finding sets that satisfy certain conditions (closed, bounded, compact). The user provides some examples and receives feedback and guidance from other forum members. The thread contains a question and partial answers from other users.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion thread from a math forum, where a user is asking for help with a problem related to metric space sets, specifically about finding sets that satisfy certain conditions (closed, bounded, compact). The user provides some examples and receives feedback and guidance from other forum members. The thread contains a question and partial answers from other users.
  
  Label: question-and-answer