Question

In: Advanced Math

1: Let X be the set of all ordered triples of 0’s and 1’s. Show that...

1: Let X be the set of all ordered triples of 0’s and 1’s. Show that X consists of 8 elements and that a metric d on X can be defined by ∀x,y ∈ X: d(x,y) := Number of places where x and y have different entries.

2: Show that the non-negativity of a metric can be deduced from only Axioms (M2), (M3), and (M4).

3: Let (X,d) be a metric space. Show that another metric D on X can be defined by ∀x,y ∈ X: D(x,y) := d(x,y)/(1 + d(x,y)).

4: Let (X,d) be a metric space.

  1. Show that every open d-ball is a d-open subset of X.
  2. Show that every closed d-ball is a d-closed subset of X.

5: Let (X,d) be a metric space. Show that a subset A of X is d-open if and only if it is the union of a (possibly empty) set of open d-balls.

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You have asked 5 questions. I have answered the first one.

Colby Messinger answered 1 year ago
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