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.
The basis for the product topology is just the product of the basis of the individual topologies. So for you just have elements of the form for . Think about what one of these looks like when you intersect it with X. That is your subspace topology.