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Math Help - Connectedness of a set: Depending on topology used?

  1. #1
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    Connectedness of a set: Depending on topology used?

    Hi dear community,
    I have a question on the subject of topology, namely the connectedness of a set.

    A set X is called connected if there are no 2 open, nonempty subsets  A,B \subset X with
     A \cup B = X .

    Now my problem is the following:
    To which topology does the term "open" refer in this definition?

    Consider, for example, the interval (0,2)  \subset \mathbb{R} .
    Then I can define the topology T(I)  := \{I, \emptyset, (0,1], (1,2)\} of open subsets for the interval I = (0,2) .

    In fact, this does give a topology, since  (0,1] \cup (1,2) = I, \\ (0,1] \cap (1,2) = \emptyset
    and so on.
    But then I =  (0,1] \cup (1,2) ,
    with (0,1] and (1,2) open in I. This would mean that I is NOT connected.

    However, using the trivial topology  T'(I) := \{I, \emptyset\} , one finds that I is in fact connected.

    The same would be true for  \mathbb{R} itself, e.g. using the trivial topology, the real numbers are connected,
    but using the discrete topology, so that every set in  \mathcal{P}(\mathbb{R}) (the power set of the real numbers) is open,
    then  \mathbb{R} is indeed NOT connected!

    However, in many math books it is mentioned that any interval I is connected, but it's not referred to the topology used.

    So is connectedness of a set really dependent on the topology used, or what the heck is my conceptual or logical error?

    Thanks in advance for help!
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    Re: Connectedness of a set: Depending on topology used?

    Quote Originally Posted by mastermind2007 View Post
    A set X is called connected if there are no 2 open, nonempty subsets  A,B \subset X with  A \cup B = X .
    Consider, for example, the interval (0,2)  \subset \mathbb{R} .
    Then I can define the topology T(I)  := \{I, \emptyset, (0,1], (1,2)\} of open subsets for the interval I = (0,2) .
    The definition you gave is for a non-connected topological space.
    The operative word there is space.
    So the example that you give is not a connected space.
    In \mathbb{R}^1 with the usual topology the set (0,2) is connected.

    Yes, the concept of disconnectedness depends upon the topology.
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    Re: Connectedness of a set: Depending on topology used?

    Hi Plato,
    thanks for the answer.

    What exactly do you mean then by the term space?
    Does the interval (0,2) not become a topological space when equipped with the topology that I mentioned?

    And what's the difference then between a connected set and a connected topological space?

    In fact, one can actually consider (0,2) as a subset of  \mathbb{R} which becomes a topological subspace with
    the subspace topology  (0,2) \cap \tau where  \tau is the topology of  \mathbb{R}.
    Then the (dis)connectedness of (0,2) would depend on the topology of  \mathbb{R}.
    Or not?
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    Re: Connectedness of a set: Depending on topology used?

    Quote Originally Posted by mastermind2007 View Post
    What exactly do you mean then by the term space?
    Does the interval (0,2) not become a topological space when equipped with the topology that I mentioned?
    And what's the difference then between a connected set and a connected topological space?
    A topological space is a pair <X,\mathcal{T}> where X is a non-empty set and \mathcal{T} is a topology on X.
    In a topological space a set is not connected if it is the union of two separated sets.
    Last edited by Plato; March 22nd 2012 at 03:29 AM.
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    Re: Connectedness of a set: Depending on topology used?

    In a topological space a set is not connected if it is not the union of two separated sets.
    You mean, it is not connected if it IS the union of two nonempty disjoint open sets.
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    Re: Connectedness of a set: Depending on topology used?

    you can think of a topology as telling you: "which points are near each other". in the discrete topology, every point is "far away" from every other point (in fact, we can define a discrete metric that makes the distance between any two distinct points as large as we like). in the indiscrete topology, all the points are "inseparable", we just have one big blob we can't look into.

    but these extreme cases are usually not what we are interested in. often, a set already comes with "some other structure". in the case of the real numbers, we have an ordered field, and we use that order and the addition to define the "usual metric" d(x,y) = |x-y|. it turns out that the open sets we get using unions of open intervals, are the same open sets we get using the metric.

    our "intuitive" notions of "connectedness" stem from properties of the standard topology on Rn. with a "non-standard" topology (such as the cofinite topology, or the sector topology), we often find that the results we get don't correspond to our intuition very well. the topological properties a topological space has, are a function of the topology, NOT "intrinsic properties of the underlying set."
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    Re: Connectedness of a set: Depending on topology used?

    our "intuitive" notions of "connectedness" stem from properties of the standard topology on Rn. with a "non-standard" topology (such as the cofinite topology, or the sector topology), we often find that the results we get don't correspond to our intuition very well. the topological properties a topological space has, are a function of the topology, NOT "intrinsic properties of the underlying set."
    Ah, okay. It's good that you mention the fact that it is not an intrinsic property of the set. There are some books that tell you actually the opposite, which made me confused.

    So it's true that, considering the interval (0,2) as itself with the topology that I gave it above, is really not connected?
    My problem's not that it's not intuitive, but I didn't know if I was wrong - just because of the fact that certain textbooks declare connectedness as some "inner property" of a set, like the Hausdorff property and compactness.
    So it would also depend on the topology I give \mathbb{R} if my (0,2) was connected with respect to its subspace topology, I guess?

    Thanks for the reference to the discrete metric and distinguishability (if that word exists ).
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