Final answer:
To prove that a countably infinite complete metric space has an infinite number of isolated points, we can use the Baire Category Theorem.
Step-by-step explanation:
To prove that a countably infinite complete metric space has an infinite number of isolated points, we can use the Baire Category Theorem.
First, we define a complete metric space as a space where every Cauchy sequence converges to a point within the space. A countably infinite space is a space that can be put into a one-to-one correspondence with the natural numbers.
Now, assume that a countably infinite complete metric space has only a finite number of isolated points. Since isolated points are points that have a neighborhood that contains no other points of the space, this means that the remaining points are clustered together. However, this contradicts the completeness of the space, as there would be a sequence of points within this cluster that converges to a point outside of the cluster. Therefore, a countably infinite complete metric space must have an infinite number of isolated points.