Question

A point xo in a metric space X is said to be isolated if xo is not in the closure of X x0}. Using the Baire Category theorem, prove that the complement of any countable set is dense in X. Use this to show that an infinite complete metric space with no isolated points is uncountable.

Answer 1

A point xo in a metric space X is said to be isolated if xo is not in the closure of X x0}. Using the Baire Category theorem,
A point to in a metric Space x is said to be isolated if to is not in the closure of the using the baire category theorem, prove that the Complement of any countable Set in dense in x. Ans: We are given that (x,d) is complete metric - Space. We know that A complete metrice Space is a set of the Second pow a cantable Subset of having no isolated point must be a set of first category. Since each Singleton {x} is non. where dense set and every countable set of the counterbale Union of its points. pow if the complement if a first Category Sot of first category, then the entire set. becomes a countable union of no-where dense sets which goes aganist - the above underlined result.

Result: I. Any countable subset of x is a lot of the first category in x if it contains no isolated point. A complete metric Space is a set of the and category. if (x,d) is countable set having no isolated point then by the result 6 it's first Category which contradics the result 6. that a complete metric Space must be second category so (x,d) must be in countable,