Can Asymptotes Be Defined as Points or Circles in Exotic Topologies?

[evanghellidis]

Could an asymptote be defined as a point, or a circle? I assume it would be in a rather exotic topology, or a very trivial one. Furthermore, can we define each of these as the other's asymptote? The points would probably have to turn into lines(functions) by virtue of an extra dimension in that topology.

What I'm basically trying to figure out is how a generic attractive force, coupled with a generic mass-like property(i.e. the points can't overlap; passive repelling force) could be defined mathematically, without using vectors. I first thought of setting x,y as each other's limit(x->y and y->x), with x\neqy, but that leads to x=y, due to the nature of the limit operation. With asymptotes, I'm guessing an infinity should show up.

Am I making any sort of sense?

 

[fresh_42]

Circular asymptotes occur in vector fields (differential equation systems). They are called attractors or repellers, depending on whether the flows rush in or come from it. The concept doesn't make much sense for usual functions.