Proving $\mathbb{R}$ is connected via path connectedness

Question

The standard proof given for the connectedness of $\mathbb{R}$ seems to me to be along the lines of Understanding the proof of "connected set is interval.".

A much simpler argument, I believe, is that $\mathbb{R}$ is pathconnected and hence connected ( open subsets of normed vector spaces are connected iff they are path-connected).

Why is the linked proof prefered ( or even stated) to this argument? Is my argument false/ less general? Many thanks.

Answer 1

The fact that path-connected implies connected, in fact relies on the fact that $[0,1]$ is connected. And if you already know $[0,1]$ is connected you have already shown that $\Bbb R$ is connected too (it follows from general facts in ordered spaces; usual proofs use that bounded sets have sups etc., and proofs apply to both $[0,1]$ and $\Bbb R$ or any interval).

So proving $\Bbb R$ connected via path-connectedness is a circular argument.

Answer 2

Not an answer to your specified question, but for a connected space $X$ is equivalent:

I) $X$ is connected

II) $\emptyset$ and $X$ are the only sets that are open and closed

III) Every continuous map of $X$ into a discrete (finite) space is constant

IV) Every continuous map $X\to\{-1,1\}$ is constant

The proof is pretty short and simple.

So in my opinion the easiest argument for $\mathbb{R}$ beeing connected, is that $\emptyset$ and $\mathbb{R}$ are the only closed and open sets.