Open connected and path connected

Question

I know each open connected space of $R^n$ where $n>1$ is path connected. I am wondering about open connected subset of $R$ is path connected. It seems to me yes , but I am not entirely sure. Any help will be appreciated.

Answer 1

Yes. Clearly, any open subset of $\Bbb R^n$ is locally pathwise connected. By Exercise 6.3.10 from Engelking’s “General topology” each connected and locally pathwise connected space is pathwise connected. This result follows from the fact that in a locally pathwise connected space a maximal locally pathwise set containing a given point is open.

Answer 2

Although the answer by Alex Ravsky is fine, you can also observe that any connected subspace $S\subset \mathbb R$ is path-connected as follows:

(1). $S$ is convex. For if $a,b\in S$ with $a<b, $ and if $a<c<b$ with $c\not \in S$ then $(-\infty,c)\cap S$ and $(c,\infty)\cap S$ are disjoint non-empty open subsets of $S$ whose union is $S.$

(2). For $a,b\in S$ and $t\in [0,1]$ let $f_{a,b}(t)=ta+(1-t)b.$ Then $f_{a,b}$ maps $[0,1]$ into $S$ because $S$ is convex.