Sheng Li (gmachine1729) wrote,
Sheng Li
gmachine1729

A nice consequence of Baire category theorem

Originally published at 狗和留美者不得入内. You can comment here or there.

In a complete metric space X, we call a point x for which \{x\} is open an isolated point. If X is countable and there are no isolated points, we can take \displaystyle\cap_{x \in X} X \setminus x = \emptyset, with each of the X \setminus x open and dense, to violate the Baire category theorem. From that, we can arrive at the proposition that in a complete metric space, no isolated points implies that the space uncountable, and similarly, that countable implies there is an isolated point.

 

Tags: analysis, real analysis
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