Is this proof for a topological basis ok?

  • #1
On the plane [tex]R^{2}[/tex] let,

B= {(a,b) x (c,d) [tex]\subset[/tex] [tex]R^{2}[/tex] | a < b, c < d }

a.) Show that B is a basis for a topology on [tex]R^{2}[/tex].


This means I have to show that every x in [tex]R^{2}[/tex] 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 [tex]\in[/tex] [tex]R^{2}[/tex] [tex]\subset[/tex] B, since B is the set of all open rectangles on [tex]R^{2}[/tex].

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

This B is a basis.

How's it look?
 

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