Question

Let X be an infinite set, for each x, y ∈ X , we define

d(x,y)= ∑
0. if x = y
1, if x≠ y

A) Show that this metric (discreet metric) is in fact a metric.

B) Show that, A⊂X, if x∈A then not necessarily x ∈ A'. That is, not all interior points are limit points.

Answer 1

The discrete metric is a metric because it satisfies the three properties. Not all interior points are limit points in a set. A counterexample can be provided to demonstrate this.

Step-by-step explanation:

Part A:

To show that the discrete metric is a metric, we need to verify the three properties of a metric:

1. Non-negativity: (x,x) = 0 for all x in X. This is true since (x,x) = 0 for any x in X.
2. Identity of indiscernibles: (x,y) = 0 if and only if x = y. This is true since (x,y) = 0 if and only if x = y.
3. Symmetry: (x,y) = (y,x) for all x, y in X. This is true since (x,y) = (y,x) for any x, y in X.

Part B:

To show that not all interior points are limit points, we can provide a counterexample. Let A be the set of all positive integers in X, and let x be any element in A. Since the discrete metric assigns a distance of 1 for distinct elements, there is no element in A within a distance of less than 1 from x. Therefore, x is not a limit point of A.