Ngô Quốc Anh

August 31, 2008

Tập hoàn hảo không chứa số hữu tỉ nào

Filed under: Các Bài Tập Nhỏ, Linh Tinh — Ngô Quốc Anh @ 15:03

Does there exist a nonempty perfect subset of that doesn’t contain any rational number?

Solution. Let be an enumeration of the rationals and let S = \bigcup_{k = 1}^\infty \left(q_k - 2^{ - k},q_k + 2^{ - k}\right). Its complement is closed, contains no rationals, and is uncountable because \mu (S) < 2\Rightarrow \mu (S') = \infty. It follows from the Cantor-Bendixson Theorem that has a nonempty perfect subset.

1 Comment »

  1. Quốc Anh thân mến,

    Bạn có thể cho mình biết cách gõ latex trên blog được k?

    Thân mến,

    Comment by dangtuanhiep — September 11, 2008 @ 8:41

RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Create a free website or blog at

%d bloggers like this: