# 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:
• #### Why Chinese do not owe anything to America for their success

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

• #### My partially successful attempt at proving the Cantor-Schroder-Bernstein theorem

Originally published at 狗和留美者不得入内. You can comment here or there. Today, I may well have learned for the first time the proof of Cantor’s…

• #### 丘成桐对知乎名为 Zeldovich Yakov 的留俄留法数学家的起诉

Originally published at 狗和留美者不得入内. You can comment here or there. 以下也有他对我的起诉。

• Post a new comment

#### Error

Anonymous comments are disabled in this journal

default userpic