“Describe the topology that {(x,0)∈ℝ^2|x∈ℝ} inherits as a subspace of ℝ^2 with the standard topology”

Question

This is a problem from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa.

PROBLEM

"Let X = {(x, 0) ∈ ℝ^2 | x ∈ ℝ}, the x-axis in the plane. Describe the topology that X inherits as a subspace of ℝ^2 with the standard topology."

THOUGHTS

The standard topology on ℝ^2 is generated by the collection of open balls.

X = {(x, 0) ∈ ℝ^2 | x ∈ ℝ} is every real number point on the x-axis in R^2.

If we let X inherit a subspace of ℝ^2, does it become the standard topology on ℝ? My thinking is that every real number point is open in R^2. If we intersect that with X, we get all the open intervals in ℝ.

If that's even correct, how can I phrase it better?

As always, I appreciate any help.

Answer 1

The inherited topology is generated by the intersection of the open balls with the real line. Since the intersections are open intervals the inherited topology is the standard topology of the real line.