Why topology is called Rubbersheet Geometry? Usually topology classes starts with comparing doughnut and tea cup. But after introductory class teacher will move to the definition of topology as a collection of subsets of a set having certain properties..
At what point does this meet with our "rubber sheet geometry"
 A: IMO, the rubber sheet analogy is really just to help you visualize a physical surface for which things like "the distance between two points" isn't really meaningful. And maybe visualizing stretched and deformed open discs might help get you used to the idea of working in terms of an open basis for a topology when previously you're only familiar with working with open discs (e.g. the $\epsilon-\delta$ definition of limit for metric spaces).
If you really want to take the rubberness somewhat more literally, you probably want to look into things like homotopy or deformations.
A: Once you have defined a certain structure (e.g. topological space, metric space, group, ring...) it is good to analyze its automorphisms.  In the case of topological spaces the automorphisms are homeomorphisms, and by studying basic examples one finds that homeomorphisms can dramatically distort what one would like to think of as the geometry of the underlying space.
For example, the map $[0,1] \to [0,2]$ given by $x \mapsto 2x$ is a homeomorphism even though it "stretches" the space and hence distorts distance.  More dramatically, the stereographic projection map $S^2 - \{point\} \to \mathbb{R}^2$ is a homeomorphism even though it stretches the space by arbitrarily large factors near the point removed and forgets curvature entirely.
Still, there are many natural invariants of homeomorphism, such as connectedness, which distinguish between spaces that are obtained from one another by "tearing".  The point is that all of these invariants (as well as the definition of homeomorphism itself) can be defined purely in terms of open sets, i.e. the axioms of topology.
(That said, the ubiquitous comparisons between coffee cups and donuts are justified by the classification of surfaces, a deep theorem in topology that you may not see in a first course.)
