(1) This is a horror story which is told to undergrads/graduate students: There was once a graduate student who was close to finishing their PhD on objects called anti-metric spaces. These are defined as follows: An anti-metric space is a pair (X, d) where X is a set and d : X R20 with the following properties. (a) d(x, y) = 0 if and only if x = y. (b) d(x, y) = d(y, x). (c) d(x, y) > d(x, z) + d(z, y). Notice that an anti-metric space is defined similarly to a metric space but with the inequality in (3) turned around. Show that if (X, d) is an anti-metric space, then X contains only 1 element. Hence, these are not exciting elements. The moral of the story is that if you are going to study an abstract object, you should have some examples. (2) Decide whether the following or not are metric spaces (a) (R,d) where d(x, y) = (x y). (b) (R, d) where d(x, y) = |x y|. = (3) Suppose that (X, dx) and (Y, dy) are metric spaces. Consider the space (XxY, dmax) where dmax ((a, b), (c, d)) : max{dx (a, c), dy (b, d)}. Prove that this is a metric. (4) Consider the metric space (0, 1) with the standard metric. Give a map T : (0, 1) (0,1) such that the resulting discrete dynamtical system has no recurrent points. Justify why your solution works. Consider the metric space R2 with the standard metric. Give a map T : R > dynamtical system has the following properties: (a) The resulting system has no periodic points. R2 such that the resulting discrete (b) There exists a global attracting point, i.e. there exists a point x = R such that for any y R, we have that limnoTn(y)=x. Hint: Try finding T which is not continuous. Justify why your solution works.