This is a crosspost from stackexchange. I'm not completely sure whether the question below is research-level, but I have not yet found an obvious answer, and what I have found thus far suggests that it might be research-level.

A topological space $X$ is *path-connected* if for points $a, b \in X$, there is a continuous function $f : [0, 1] \to X$ such that $f(0) = a$ and $f(1) = b$. This condition can be strengthened to *arcwise-connectedness*, which additionally requires that $f$ is a topological embedding, or homeomorphic onto its image. Clearly if $X$ is arcwise-connected, then it is also path-connected. If $X$ is Hausdorff, then the converse also holds. There is a not-so-trivial proof of the converse in Chapter 31 of Willard's *General Topology* on Peano spaces (there is also some related discussion on nlab).

A space is called *$n$-connected* if all its homotopy groups vanish up to $n$, or $\pi_i(X) = 0$ for all $i \le n$. Equivalently, $X$ is $n$-connected for any $i \le n$, and continuous function $f : S^i \to X$, we can extend it to a continuous map $F : D^{n+1} \to X$. This generalizes path-connectedness, it is the same as $0$-connectedness.

Can the "path-connectedness implies arcwise-connectedness" property be generalized to higher connectivity? That is, suppose we say that $X$ is "*$n$-arcwise-connected*" if for any $i \le n$ and embedding $f : S^i \to X$, there is an embedding $F : D^{i+1} \to X$ extending $f$. Are there sufficient conditions on $X$ (like $X$ is Hausdorff) for which we have $n$-connectedness implying $n$-arcwise-connectedness? Are there known results on this?

My motivation is: given an embedding $S^i \to X$ in an $n$-connected space, I want to extend it to an embedding $D^{i+1} \to X$. $n$-connectedness alone doesn't immediately give me this.

path-connected, I think you should give a definition for this lesser known concept. $\endgroup$ – Jean Duchon Apr 23 '18 at 9:46