Path-connectedness and connectedness of $\mathbb{R}$

Question

I had a thought about the path-connectedness of $\mathbb{R}$. The standard way of showing the path-connectedness of $\mathbb{R}$ can be that it is both open and connected and so must be path-connected too.

But I had thought about another argument, say $a$, $b\in\mathbb{R}$, can we just define a path that connects $a$ and $b$ as follows? $\gamma:[0, 1]\to \mathbb{R}$ such that $\gamma(t)=a+(b-a)t$ which precisely take $a$ to $b$ and the image precisely lie in $\mathbb{R}$ as well since $\mathbb{R}$ is closed under multiplication and addition?

Obviously, the first method sounds much more rigorous but it will be much longer to prove so I was wondering if the second method is insufficient in proving the path-connectedness?

Answer 1

Actually, your method is perfectly sound and rigorous. It is actually the natural way of doing it. And, in fact what you descrined as the “standard way” of proving it doesn't make sense. By that argument, and sense every topological space is an open subset of itself, every connected topological space would be path-connected true, which is not true.

Answer 2

I would say that the standard method to show that $\mathbb{R}$ is path connected is actually the second one... and then showing that it is connected since it is path connected.

Answer 3

First one shows that in a linearly ordered space topological space $X$ (as $\Bbb R$ is), $X$ is connected iff it has no gaps and no jumps. As $\Bbb R$ has a dense order, it has no jumps and as it has the lub-property, it has no gaps. It follows that $\Bbb R$ and $[0,1]$ are both connected.

That $\Bbb R$ is then path-connected follows from your second argument: we can find a path between any two points by using the continuous addition and multiplication, and this argument works in any $\Bbb R^n$ or topological vector space over $\Bbb R$. Paths are defined on subsets of $\Bbb R$ and so that works really well. Path-connectedness implies connectedness just because we know that $[0,1]$ is already connected. So connectedness of intervals is the most basic fact upon which we rely.