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Graphs of Continuous Functions and the Subspace Topology

  1. Mar 22, 2013 #1
    Let U be a subset of ℝn be an open subset and let f:U→ℝk be a continuous function.

    the graph of f is the subset ℝn × ℝk defined by

    G(f) = {(x,y) in ℝn × ℝk : x in U and y=f(x)}

    with the subspace topology

    so I'm really just trying to understand that last part of this definition.

    If we let X = G(f), and S is a subset of X, we define the subset topology on S by saying some subset U of S to be open in S iff there exists an open subset V of X s.t. U=V and S.

    not sure how to really apply this definition in this problem. Any help?
     
  2. jcsd
  3. Mar 22, 2013 #2
    A low-dimension example may help. Suppose the graph of f is a sphere in R3 where R3 has the standard topology. The subspace topology of G(f) is defined by intersecting sets that are open in the standard topology of R3 with G(f). That is, a subset of G(f) is open in the subspace topology if it is the union of some collection of open balls in R3 intersected with the sphere. Ie., we expect the curved open disc that is formed by intersecting a single open ball with the sphere to be open in the topology of the sphere.
    In your definition, they assume RnxRk is already equipped with a topology, and they want the graph G(f) to inherit that topology as described above through the use of intersections.
     
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