How are topologically separated graphs related in linear equations?

[trambolin]

Hi there,

I am kind of blocked by the "topologically seperated" phrase in the following sense. The reading comes to the paragpraph as such

"... Relations are topologically seperated, loosely speaking, if the distance from points of one relation to the other grows without bound..."

Anyone can shed some light on this? Especially if we have two graphs :

$$\begin{array}{l} G_1 = \left\{ {\left( {\begin{array}{*{20}c} x \\ y \\ \end{array}} \right):x = Ay} \right\} \\ G_2^- = \left\{ {\left( {\begin{array}{*{20}c} x \\ y \\ \end{array}} \right):y = Bx} \right\} \\ \end{array}$$

It is said that "bla bla bla ... if and only if the graph of A and the inverse graph of B are topologically separated i.e. $G_1 \cap G_2^- = \{0\}$" Let's keep A,B linear for now. I don't get how come the two sentences are related in a topological sense.

Thanks

 

[fresh_42]

They are not related in a topological sense, as long as you do not define a topology. Or as in this case, a metric, since you talk about distances.

The little that can be said from the "loosely speaking is, that $A$ and $B$ in your example define linear functions of different slope, so that the points of the graphs get further apart the bigger the distance to $(0,0)$ is. Loosely questions can only get loosely answers.