How do you cut a space?

By cutting a line the only thing you can get is line segment. As much you cut, you never get to anything like a point. But, how do you cut a line if not at specific point? If there are no points, then there can be no line segments either.

By cutting a plane, the only thing you can get is plane segments. As much you cut, you never get to anything like a line (or curve). But how do you cut a plane, if there is no lines? If there are no lines, there are no plane segments.

Same goes for space.