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 $\displaystyle \mathbb{R}^2$ you just have elements of the form $\displaystyle (a,b) \times (c,d)$ for $\displaystyle a,b,c,d \in \mathbb{R}$. Think about what one of these looks like when you intersect it with X. That is your subspace topology.