Does this generalization of path-connectedness also cover general connectedness?
I've got the following idea to generalise path-connectedness:
Given a topological space $P$ and a subspace $S$, a space $X$ is $(P,S)$-connected if every continuous function $f:S\to X$ can be extended to a continuous function $g:P\to X$.
The usual path-connectedness is obtained using $P=[0,1]$ with the usual topology and $S=\{0,1\}$. Since every function $f:\{0,1\}\to X$ is continuous, $f$ just picks two arbitrary points, and then $g$ is a path between those points.
Using the closed unit disc as $P$ and the unit circle as $S$ one also obtains whether all path-connected components are simply connected.
Now I wonder if there is also a choice of $P$ and $S$ that recovers general connectedness, that is the nonexistence of clopen sets other than the empty set and the full space.
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