I was wondering if someone could help with this question. I am stuck on part of the question showing that the following set is a regular surface in $\displaystyle \mathbb{R}^3$

$\displaystyle S =\{ p + uX + vY \mid u,v \in \mathbb{R} \}, $i.e. the affine plane through a point p spanned by linearly independent vectors.

In particular I am trying to show that the function $\displaystyle f: \mathbb{R}^2 \rightarrow S \cap \mathbb{R}^3 = S $, given by $\displaystyle f(x,y) = p + xX + yY $ is a homeomorphism. I think its clear that it is bijective and continuous but I am having difficulty showing that it's inverse is continuous, that is for some open set O $\displaystyle \subseteq \mathbb{R}^2$,

$\displaystyle f(O) = S \cap E$ for some open set E in $\displaystyle \mathbb{R}^3$. How do I find out what E must be?

any help would be appreciated