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Math Help - metric space sets

  1. #1
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    metric space sets

    Hi,

    Can one please help me with the following game:

    metric space sets-untitled.jpg

    I just can not find a set.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    A little help:

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

    Choose A=\{0\}

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

    A is compact iff A is closed and bounded.

    ...

    Fernando Revilla
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  3. #3
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    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. [0,1] is closed, bounded, and compact.
    (0,\infty) is not closed, not bounded, and not compact.
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  4. #4
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    Quote Originally Posted by Plato View Post
    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. [0,1] is closed, bounded, and compact.
    (0,\infty) is not closed, not bounded, and not compact.

    Hi sorry about this.

    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
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  5. #5
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    I don't understand what the question asks about the different possibilites
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  6. #6
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    Quote Originally Posted by 1234567 View Post
    I don't understand what the question asks about the different possibilites
    Think of a truth-table
    \begin{array}{*{20}c} {closed} &\vline &  {bounded} &\vline &  {compact}  \\\hline<br />
   Y &\vline &  Y &\vline &  Y  \\<br />
   Y &\vline &  Y &\vline &  N  \\<br />
   Y &\vline &  N &\vline &  Y  \\<br />
   Y &\vline &  N &\vline &  N  \\<br />
   N &\vline &  Y &\vline &  Y  \\<br />
   N &\vline &  Y &\vline &  N  \\<br />
   N &\vline &  N &\vline &  Y  \\<br />
   N &\vline &  N &\vline &  N  \\ \end{array}
    I gave you the answers for the first line and the last line.
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  7. #7
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    Quote Originally Posted by Plato View Post
    Think of a truth-table
    \begin{array}{*{20}c} {closed} &\vline &  {bounded} &\vline &  {compact}  \\\hline<br />
   Y &\vline &  Y &\vline &  Y  \\<br />
   Y &\vline &  Y &\vline &  N  \\<br />
   Y &\vline &  N &\vline &  Y  \\<br />
   Y &\vline &  N &\vline &  N  \\<br />
   N &\vline &  Y &\vline &  Y  \\<br />
   N &\vline &  Y &\vline &  N  \\<br />
   N &\vline &  N &\vline &  Y  \\<br />
   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
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