Is the following definition of objects (defined by compact sets) which may be 'pulled apart' without physically touching each other acceptable?
Two compact sets $A, B \subset \mathbb{R}^3$ are said to be separable if there exists positive $b\in\mathbb{R}$ such that there exists a continuous map $\mathbf{f}:A \times \mathbb{R}\rightarrow \mathbb{R}^3$ where $\mathbf{f}(\mathbf{x},t)=R(t)\mathbf{x}+\mathbf{d}(t) \notin B$ for all $t\in[0,b]$ and $\mathbf{x}\in A$ and $\mathbf{f}(\mathbf{x},t)\notin \text{conv}(B)$ for all $t>b$ and $\mathbf{x}\in A$ ($R$ is a proper rotation matrix whose entries change as a function of $t$ and conv() is the convex hull operat0r).
Because I do not know much about differential geometry, topology, or analysis, edits to this post's title and body that use more 'correct' terminology are appreciated.
If this definition is true, does it also hold for sets in $\mathbb{R}^3 $ that are only bounded?