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Metric Spaces: Exercise 1.14 from Introduction to Topology (Dover)

  1. Feb 16, 2013 #1
    1. The problem statement, all variables and given/known data
    X is a metric and E is a subspace of X (E[itex]\subset[/itex]X)
    The boundary ∂E of a set E is defined to be the set of points adherent to both E and the complement of E,

    ∂E=[itex]\overline{E}[/itex][itex]\cap[/itex]([itex]\overline{X\E}[/itex])
    (ignore the red color, i can't get it out)

    Show that E is open if and only if E[itex]\cap[/itex]∂E is empty. Show that E is closed if and only if ∂E [itex]\subseteq[/itex] E

    2. Relevant equations

    ∂E=[itex]\overline{E}[/itex][itex]\cap[/itex]([itex]\overline{X\E}[/itex])
    (ignore the red color, i can't get it out)

    3. The attempt at a solution

    To begin with I don't understand the equation because it seems to me that ([itex]\overline{X\E}[/itex])=E , so,
    ∂E = [itex]\overline{E}[/itex][itex]\cap[/itex]([itex]\overline{X\E}[/itex]) = [itex]\overline{E}[/itex][itex]\cap[/itex]E= empty set

    Can anyone explain this to me?
     
  2. jcsd
  3. Feb 16, 2013 #2

    micromass

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    There is no need to make a separate itex-tag for each math symbol. Things like this

    Code (Text):

    [itex]E\cap \overline{X\setminus E}[/itex]
     
    work fine.
     
  4. Feb 16, 2013 #3

    Dick

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    You are probably thinking the bar means complement. It doesn't. It means closure. Look up the definition and using it explain why you think ##\overline{X\backslash E}=E##. For the red tex problem use \backslash instead of \.
     
  5. Feb 16, 2013 #4
    Thanks a lot guys! Without you I couldn't do my self-study!
     
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