I suspect I will show my ignorance here, but this 'theorem' I would consider to be intuitively sensible, but I cannot find anything similar by looking through a few books or on the web. If would seem true in principal, but it probably needs some modification to how I have formulated it below. I was wondering if anyone know where I might find a proof of such a thing.

Let $M$ be a connected manifold of dimension $>n$ . Let $f:\mathbb{S}^{n} \rightarrow M$ be a map that is a homeomorphism onto its image $C=f(\mathbb{S}^{n})$ . Then

(1)- if $M$ has dimension $n+1$, then $M-C$ is the disjoint union of two open sets $A,B$ , each of which is path connected.

(2)- if $M$ has dimension $\geq n+2$ , then $M-C$ is path connected.

Notice that in (1) the usual Jordan curve theorem would say that $A$ is bounded and $B$ is unbounded, but this wouldn't seem to hold in the generalised case.

My motivation for the above is that it would give a nice way to show that a $S^2$ is not homeomorphic to the disk $D^3$.