Graphs of Continuous Functions and the Subspace Topology

[BrainHurts]

Let U be a subset of ℝⁿ be an open subset and let f:U→ℝ^k be a continuous function.

the graph of f is the subset ℝⁿ × ℝ^k defined by

G(f) = {(x,y) in ℝⁿ × ℝ^k : 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?

 

[slider142]

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 RⁿxR^k 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.