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Elementary Topology

  1. Sep 3, 2009 #1
    1. The problem statement, all variables and given/known data

    Determine if the set of points (x,y) on y = |x-2| + 3 - x are bounded/unbounded, closed/open, connected/disconnect and what it's boundary consist of.

    2. Relevant equations

    3. The attempt at a solution

    I know that the set is closed, and then by definition of a closed set it's boundary is itself. As far as bounded/unbounded goes, it seems unbounded when I graph it because I cannot see the entire graph. I'm unsure about connectedness and do not know how to determine it.

    Any help is appreciated.
     
  2. jcsd
  3. Sep 3, 2009 #2

    Dick

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    Well, in (x,y) x is certainly not bounded. It runs from -infinity to infinity. To think about connectedness, do you see that the function is continuous? That means its graph is a continuous curve with no breaks in it, right? What's the definition of 'connected' that you are using?
     
  4. Sep 3, 2009 #3
    Just so you know, that's not true in general. A closed set contains its boundary. For example, the unit 2-ball [tex](x^{2} + y^{2})^{1/2} \leq 1[/tex] is closed but is not the same as its boundary which is the 1-sphere [tex](x^{2} + y^{2})^{1/2} = 1[/tex]. I could just be being pedantic though and you could well have known that and just not felt like spelling it out.
     
  5. Sep 3, 2009 #4
    My guess then would that it is disconnected because of the absolute value in the function.
     
  6. Sep 3, 2009 #5

    Dick

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    Guess?? Why are you guessing?? I'll ask you once more. What's the definition of a connected set?
     
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