Question

Suppose that X is a non-empty, complete, and countable metric space. Use Baire Category Theorem only to prove directly that X has an isolated point. Include a precise statement of the theorem in your proof and indicate how it is being applied.

Answer 1

Let d be metric defined on the non empty set X which is countable and the metric space is complete.

Want to prove that: X has an isolated point.

We will prove it by the method of contradiction.

Now suppose X doesn't have any isolated point, which means every point is a limit point.

Now fix an enumeration

.

For any arbitrary let

,

Since every singleton set is closed, and complement of any closed set is an open set. Thus is an open set in X.

As we know every point of X is limit and adding or subtracting a single point doesn't affect on the limit points.

Therefore Closure set of is the whole X, which means is dense in X.

Now for each is open and dense in X.

Now consider the Baire category theorem, which states that

In a non-empty complete metric space X, intersection of countable family of open dense subset is dense in X.

Therefore using the above theorem:

is dense in X.

Now for any using the definition

Thus no elements of X, belongs to therefore

is an empty set which can not be a dense set.

This leads to a contradiction, that is, our assumption is wrong.

Therefore X has an isolated point.