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Math Help - Abstract simplicial complexes?

  1. #1
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    Abstract simplicial complexes?

    I'm trying to learn algebraic topology on my own and I'm currently reading Lee's book "Introduction to Topological Manifolds". But I'm stuck on the definition of abstract simplicial complexes.

    Quote:
    "We define an abstract simplicial complex to be a collection K of nonempty finite sets called (abstract) simplices, subject only to one condition: If s is in K, then every nonempty subset of s is in K."

    I try to relate this to the concept of an euclidean simplicial complex, but it doesn't make any sense to me.

    For example:
    Consider the 1-simplex spanned by two vertices (0) and (1). The 1-simplex is then the closed interval [0,1].
    Let K be the euclidean simplicial complex with sets {(0), (1) and [0,1]}, i.e. with three elements.
    Om ok with this so far, but then K fails to be an abstract simplicial complex since the interval (1/3, 2/3) is not in the set.
    (The definition clearly states that (1/3, 2/3) should be in the set as it is a nonempty subset of [0,1])
    This seems odd as my intuition tells me that abstract simplicial complexes should be a generalization of euclidean simplicial complexes and this would thus be a counterexample.

    I understand that some part of my reasoning must be wrong, but I cannot see in what way.

    Can anyone please help me and try to explain?
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  2. #2
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    Quote Originally Posted by patrik1982 View Post
    I'm trying to learn algebraic topology on my own and I'm currently reading Lee's book "Introduction to Topological Manifolds". But I'm stuck on the definition of abstract simplicial complexes.

    Quote:
    "We define an abstract simplicial complex to be a collection K of nonempty finite sets called (abstract) simplices, subject only to one condition: If s is in K, then every nonempty subset of s is in K."

    I try to relate this to the concept of an euclidean simplicial complex, but it doesn't make any sense to me.

    For example:
    Consider the 1-simplex spanned by two vertices (0) and (1). The 1-simplex is then the closed interval [0,1].
    Let K be the euclidean simplicial complex with sets {(0), (1) and [0,1]}, i.e. with three elements.
    Om ok with this so far, but then K fails to be an abstract simplicial complex since the interval (1/3, 2/3) is not in the set.
    (The definition clearly states that (1/3, 2/3) should be in the set as it is a nonempty subset of [0,1])
    This seems odd as my intuition tells me that abstract simplicial complexes should be a generalization of euclidean simplicial complexes and this would thus be a counterexample.

    I understand that some part of my reasoning must be wrong, but I cannot see in what way.

    Can anyone please help me and try to explain?
    An abstract simplicial complex S and its geometric realization of S are two different things.

    Label each vertex as (an abstract) 0-simplex {a} and {b}, respectively. Then your 1-simplex becomes {a,b}.

    Now your abstract simplicial complex S is S={{a}, {b}, {a,b}}, which satisfies your condition.
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  3. #3
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    Quote Originally Posted by TheArtofSymmetry View Post
    An abstract simplicial complex S and its geometric realization of S are two different things.

    Label each vertex as (an abstract) 0-simplex {a} and {b}, respectively. Then your 1-simplex becomes {a,b}.

    Now your abstract simplicial complex S is S={{a}, {b}, {a,b}}, which satisfies your condition.
    Ok, so a simplex is then simply a collection of vertices (may if be none, one, two or many vertices)... That perfectly makes sense now when I'm reading the text again.

    So simple now when I get it, I just needed it rephrased in a good way.

    Thank you!
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