I am interested in learning what is behind the following idea, which I'm sure must have been thought about before (if it is correct). It's pretty handwavy, but here are a few examples.
First start in $\mathbb{R}$. The only hyperplane is the $0$-plane, i.e. the origin, and we can surround it by $S^0$, just two points on either side of the origin.
Then in $\mathbb{R}^2$, the $0$-plane can be surrounded by $S^1$, and a $1$-plane, a line, can be "surrounded" by $S^0$, in the sense that on both sides of the line, we can place a point.
In $\mathbb{R}^3$, we similarly have that the origin is surrounded by $S^2$, a line is surrounded (encircled) by a circle, $S^1$, and a plane is "surrounded" by two points, $S^0$.
Continuing the pattern in higher dimension without being able to visualize it, my question is
Do we have that in $\mathbb{R}^n$, $S^k$ "surrounds" an $(n-k-1)$-plane? (Here I would need a more proper definition of surrounding instead of relying on intuition. What would be this definition?)
Perhaps this is just a way of saying that the complement of an $(n-k-1)$-plane in $\mathbb{R}^n$ is homotopy equivalent to $S^k$, if this is true?