# A simple question about Theorem 2.47 on p.42 in “Principles of Mathematical Analysis 3rd Edition” by Walter Rudin.

I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin.

There is the following theorem on p.42:

Theorem 2.47
A subset $$E$$ of the real line $$\mathbb{R}^1$$ is connected if and only if it has the following property: If $$x \in E, y \in E$$, and $$x < z < y$$, then $$z \in E$$.

Rudin didn't write the following more concrete result.

Why?

I wish if Rudin had written the following more concrete result:

A subset $$E$$ of the real line $$\mathbb{R}^1$$ has the following property: If $$x \in E, y \in E$$, and $$x < z < y$$, then $$z \in E$$

if and only if

(1) $$E = (a, b)$$ for $$a, b \in \mathbb{R}$$ such that $$a \leq b$$ or
(2) $$E = [a, b]$$ for $$a, b \in \mathbb{R}$$ such that $$a \leq b$$ or
(3) $$E = [a, b)$$ for $$a, b \in \mathbb{R}$$ such that $$a \leq b$$ or
(4) $$E = (a, b]$$ for $$a, b \in \mathbb{R}$$ such that $$a \leq b$$ or
(5) $$E = (a, +\infty)$$ for $$a \in \mathbb{R}$$ or
(6) $$E = [a, +\infty)$$ for $$a \in \mathbb{R}$$ or
(7) $$E = (-\infty, b)$$ for $$b \in \mathbb{R}$$ or
(8) $$E = (-\infty, b]$$ for $$b \in \mathbb{R}$$ or
(9) $$E = (-\infty, +\infty)$$.

• Your theorem has nothing to do with connectedness. It complements Rudin's theorem but it is not a replacement. – Kavi Rama Murthy Feb 4 at 9:15