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Mbedding in topology

  1. Feb 23, 2009 #1
    x0 [tex]\in[/tex]X and y0[tex]\in[/tex]Y,
    f:X[tex]\rightarrow[/tex]X x Y and g: Y[tex]\rightarrow[/tex]X x Y defined by
    f(x)= x x y0 and g(y)=x0 x y are embeddings

    This is all I have...
    f(x): {(x,y): x[tex]\in[/tex]X and y[tex]\in[/tex]Y}
    g(y): {(x,y): x[tex]\in[/tex]X and y[tex]\in[/tex]Y}

    right?
    soo... embeddings are.... one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.


    i don't know how to do go about this....
     
  2. jcsd
  3. Feb 24, 2009 #2

    CompuChip

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    An embedding is literally what it says: a function that embeds one topological space in another. For example, if n > 1 then
    [tex]f: \mathbb{R} \to \mathbb{R}^n, x \mapsto (x, 0, 0, \cdots, 0)[/tex]
    embeds the real line in some space (like the plane or three-dimensional Euclidean space),
    [tex]g: S^1 \to \mathbb{R}^2, \theta \mapsto (\cos\theta, \sin\theta)[/tex]
    embeds the circle in the plane, but you can also embed it in a sphere
    [tex]h: S^1 \to S^2, \theta \mapsto (\cos\theta, \sin\theta, 0)[/tex]
    which maps the circle S^1 to the equator of the sphere S^2 (you can even compose this with any rotation, which just maps it to some other great circle on the sphere).

    The function you gave, for example
    [tex]X \to X \times Y, x \mapsto (x, y_0)[/tex]
    is a simple example of a general embedding. Basically what you do is choose a fixed point in Y, and then embed X in X x Y by simply filling in the "missing" coordinates with y0. For example, the embedding of the real line in Euclidean space which I called f above, can be obtained in this way: set [itex]X = \mathbb{R}, Y = \mathbb{R}^{n - 1}, y_0 = \vec 0_{n-1}[/itex] where [itex]\vec 0_{n-1}[/itex] is the zero vector in [itex]\mathbb{R}^{n-1}[/itex].
     
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