When authors refer to arc-wise connectivity, do they mean path-wise connectivity?
I am studying space filling curves and when reading books, I either come across the concept of arc-wise connectivity or path-wise connectivity.
Can arc-wise connectivity and path-wise connectivity be taken to mean the same thing?
Let me first review what I would consider to be the standard definitions of these terms. A space $X$ is path-connected if for any $x,y\in X$ there is a continuous map $f:[0,1]\to X$ such that $f(0)=x$ and $f(1)=y$. A space $X$ is arc-connected if for any $x,y\in X$ there is an injective continuous map $f:[0,1]\to X$ such that $f(0)=x$ and $f(1)=y$.
That said, I have also occasionally seen people that treat them as synonymous, with the definition I gave for "path-connected" applying to both. Note that for reasonably nice spaces, the two notions are equivalent. In particular, any Hausdorff path-connected space is automatically arc-connected (this is fairly complicated to prove; see Does path-connected imply simple path-connected?, for instance). So if you are only thinking about Hausdorff spaces, there is no need to distinguish them.
