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How is 0-cell is a vertex

  1. Feb 23, 2010 #1
    Definition: (open cell). Let X be a Hausdorff space. A set c ⊂ X is an open
    k − cell if it is homeomorphic to the interior of the open k-dimensional ball
    Dk = {x ∈ Rk | x < 1}. The number k is unique by the invariance of
    domain theorem, and is called dimension of c.
    A 0-cell, 1-cell, 2-cell and 3-cell are called a vertex, edge, face and volume

    I am confused, what is the meaning of 0-cell and 1-cell. I can imagine a circle and a sphere without borders which resemble 2-cell and 3-cell. But how is vertex and lines are homeomorphic to D0 and D1 respectively. and how is the vertex is 0-cell and edge is 1-cell. I simply can not imagine that.
  2. jcsd
  3. Feb 23, 2010 #2


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    By definition, R^0 is the point {0}. So a 0-cell is {0}. If you plug k=1 in the definition of open k-dimensional ball, you get that D^1= {x ∈ R | |x| < 1}. That's the interval (-1,1).
  4. Feb 24, 2010 #3
    Thank you, I get it
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