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A question on Topology

  1. Mar 28, 2008 #1
    1. The problem statement, all variables and given/known data

    Let X be an ordered set. If Y is a proper subset of X that is convex in X, does it follow that Y is an interval or a ray in X?

    3. The attempt at a solution

    I considered it to be yes.

    Since in the ordinary situation, the assertion is obviously valid: check out the real line or the complex plane with dictionary order, in case of Y is not empty.

    But I wonder if it holds when Y is an empty set or a set with only a single point.
    And besides, I'm not quite sure with my own judgement, since I didn't think of any special situation (if any).
     
  2. jcsd
  3. Mar 28, 2008 #2

    morphism

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    Start by looking at the definitions of "interval" and "ray" in X. Then look at the definition of Y being convex in X; does this imply it's an interval or a ray?
     
  4. Mar 28, 2008 #3

    HallsofIvy

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    The "X is an ordered set" is important here! That should certainly be considered in the definition of "interval" and "ray".
     
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