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I was reading the definition of dimension from the book: "Topology", Munkres, 2nd ed.

Surely I don't understand, but I wonder how ℝ

^{2}can have dimension 2.

Take the open sets [tex] U_n=\{(x,y)\mid -\infty < x <\infty, n-1<y<n+1\} [/tex] for every integer n. It covers the plane but its order is 2, so the dimension should be less than 2.

Shouldn't be the difinition with balls or "squares"?

Thank you.