# Is this proof for a topological basis ok?

On the plane $$R^{2}$$ let,

B= {(a,b) x (c,d) $$\subset$$ $$R^{2}$$ | a < b, c < d }

a.) Show that B is a basis for a topology on $$R^{2}$$.

This means I have to show that every x in $$R^{2}$$ is contained in a basis element, and that every point in the intersection of two basis elements is contained in a basis element in that intersection.

So:

Each x $$\in$$ $$R^{2}$$ $$\subset$$ B, since B is the set of all open rectangles on $$R^{2}$$.

If x $$\in$$ B1$$\bigcap$$B2, then x $$\in$$ B3 $$\subseteq$$ B1$$\bigcap$$B2, since the intersection of open rectangles is also an open rectangle.

This B is a basis.

How's it look?