0 1 is an open subset of R but not of R2 when we think of R

(0, 1) is an open subset of R but not of R^2, when we think of R as the x-axis in R^2. Prove this.

Solution

Since its well known that (0,1) is open in R, so We\'ll show (0,1) is not open in R^2. Since we\'re thinking of R as the x-axis in R^2, we really mean that (0,1)x{0} is not open in R^2.

Consider the point p = (1/2,0) in (0,1)x{0}. Suppose (to get a contradiction) that (0,1)x{0} is open. Then there must be a radius r > 0 such that the open ball B(p,r) is contained in (0,1)x{0}. Now notice that the point q = (1/2,r/2) is in B(p,r) (since d(p,q) = r/2 < r) but q is not in (0,1)x{0} (since r/2 is nonzero). So that contradicts the fact that B(p,r) is supposed to be contained in (0,1)x{0}, which means it could not be open