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Math Help - quasiorders

  1. #1
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    quasiorders

    a quasiorder being defined as reflexive and transitive - not antisymmetric (if also antisymmetric then it is a partial ordering).

    it is a trivial quasiorder iff: (Ax,y)(x<=y). can someone provide me with a few canonical, if not pathological, examples of a trivial quasiorder?

    what is the relation between a neighborhood and an open set? a specialisation quasiorder .=. (x<=y) iff every neighborhood of x is a neighborhood of y. i need a few examples of a specialisation quasiorder.

    why is a singleton necessarily a closed set -- why can't it be declared open?

    thanks.
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  2. #2
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    Quote Originally Posted by nweissma
    a quasiorder being defined as reflexive and transitive - not antisymmetric (if also antisymmetric then it is a partial ordering).

    it is a trivial quasiorder iff: (Ax,y)(x<=y). can someone provide me with a few canonical, if not pathological, examples of a trivial quasiorder?

    what is the relation between a neighborhood and an open set? a specialisation quasiorder .=. (x<=y) iff every neighborhood of x is a neighborhood of y. i need a few examples of a specialisation quasiorder.

    why is a singleton necessarily a closed set -- why can't it be declared open?

    thanks.
    It's hard to think of a non-trivial example of trivial quasiorder. Define \equiv by \forall x,y, x \equiv y. It's trivially trivial, but what you gonna do?

    A neighborhood of a point x is any open set containing x. I've never heard of a specialization quasiorder.

    There are valid topologies where a singleton is an open set. One example is the discrete topology where every set is an open set. It's a theorem that a topology satisfies the T_1 separation axiom iff every set containing a single point is closed.
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  3. #3
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    neighborhoods vs open sets

    A neighborhood of a point x is any open set containing x.

    this is a mere clinical definition.

    consider: separations axioms are defined in terms of open sets. can they be defined ito neighborhoods? how?
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