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Clopen sets

  1. May 26, 2009 #1
    This isn't really a homework question, can someone just explain this bit from wikipedia?

    consider the space X which consists of the union of the two intervals [0,1] and [2,3]. The topology on X is inherited as the subspace topology from the ordinary topology on the real line R. In X, the set [0,1] is clopen, as is the set [2,3]. This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.

    and later:

    A set is clopen if and only if its boundary is empty.

    Ok...so take the set [0,1] C X where X = [o,1]U[2,3]....how is the boundary of [0,1] empty? Isn't the boundary of [0,1] the 2 points 0 and 1? So I don't really get how [0,1] is clopen in this case
     
  2. jcsd
  3. May 26, 2009 #2

    matt grime

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    What is the definition of boundary? Remember this is in the subspace topology and you shouldn't just think that your intuition about [0,1] being a subset of R is correct - after all [0,1] is open and closed...
     
  4. May 26, 2009 #3

    quasar987

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    Is 0 really in the boundary of [0,1]? By definition, it is so if every open set U of X containing 0 contains points of [0,1] and of X\[0,1]=[2,3]. Well, take for instance the open set (-1,1)nX=[0,1). It does not contain points of [2,3], so 0 is not in the boundary of X.

    What happens here is that [0,1] has boundary {0,1} in R, but not in X.
     
  5. May 26, 2009 #4
    Ah ok, thanks guys :) its more clear now
     
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