Suppose that we define a function d that maps R2 to R by d(x,y) = { 0,. if x = y 1, if x is not equal to y }. a. Prove that d is a metric on R. b. Prove that for any a in R, {a} is an open set under the metric d. c. Let X = R with the usual Euclidean distance and Y = R with the metric d defined above. Prove that f maps X to Y defined by f(x) = x is not continuous.

Please solve the Question.2. ..It is from Real Analysis.

Indicate whether the following statement is true or false? 1. If S = span{u1, U2, Uz}, then dim(S) = 3.

Which of the following statements is/are not true about non-Euclidean geometries? I. In Lobachevskian geometry, there are no parallel geodesics.  II. In Riemannian geometry, at least 2 lines are parallel to a given line. III. In non-Euclidean geometry, there exists exactly one line parallel to a given line.

q For each of the following functions diR² AR [0,00] wether d is a metric, Briefly explain say reasoning. your 0.) dixiy ) = { 0° ₂ 6.) (x,y) = (x₁+y₁| + | x₂ + yzl if x=y 2 elsewhere x=y

show that iR A is any su bret of X with the discre te metric then

For what value(s) of h is y in the span of ?1 and ?2?

EC. Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric).i. Explain why every horizontal line of the form [a,b] × {y} is compact.ii. Given any open cover Oλ of R, conclude that each line of the form [a,b] × {y} is covered by a finite subfamily of the Oλ's.iii. Using ii, show that R is compact.Hint: In iii, it may help to look at, for each horizontal line, the union of the finite subcover. How "thick" or "thin" can this union be?Note: Using the same argument as that for ℝ, it follows from this problem that a subset of ℝ2 is compact if and only if it is closed and bounded.