1: Let X be the set of all orderedtriples of 0’s and 1’s. Show that X consists of 8 elementsand that a metric d on X can be defined by?x,y ? X:d(x,y) := Number of places wherex and y have different entries.
2: Show that the non-negativity of a metric canbe deduced from only Axioms (M2), (M3), and (M4).
3: Let (X,d) be a metricspace. Show that another metric D on X can bedefined by ?x,y ? X:D(x,y) :=d(x,y)/(1 +d(x,y)).
4: Let (X,d) be a metricspace.
- Show that every open d-ball is a d-opensubset of X.
- Show that every closed d-ball is a d-closedsubset of X.
5: Let (X,d) be a metricspace. Show that a subset A of X isd-open if and only if it is the union of a (possiblyempty) set of open d-balls.
