Suppose (L_{n}) is a sequence of loops in a torus S^{1} × S^{1} converging in the Hausdorff metric to some set L in the torus. Suppose also that for each loop L_{n} the projection map p:S^{1} × S^{1} > S^{1} defined by p(x,y) =x when restricted to L_{n} is not nullhomotopic, then can we conclude that the restriction of the projection map p to L is also not null homotopic? Or are there counter examples?
Your intuition is correct: the map p : L → S^{1} is not nullhomotopic. If p : L → S^{1} was nullhomotopic, then it would factor through the universal cover ℝ of S^{1}. Let C_{0} ⊂ ℝ × S^{1} be a loop in that
separates L_{(0)} from L_{(1)}. The projection C ⊂ S^{1} × S^{1} of C_{0} is then a loop in the complement of L, Since L lies in the complement of C and L_{n} → L in the Hausdorff metric, 


EDITED. The following example should not be conisdered as a contreexample, it is just a well known example of a pathology that can happen. EXAMPLE. $L$ is composed of a union of the vertical circle $(0, S^1)$ and a disjoint $R^1$ that is emdedded in $T^2$ in such a way, that it accumulates to $(0, S^1)$ from both sides. This $R^1$ projects oneto one to the horisontal circle without a point and can be represented a as graph (function of $x$), than if is given by the following formula: $y= (sin(1/x))$ The point is that such $L$ can be Hausodrff approximated by a sequence of circles $s_n$ that are not nullhomotopic ($s_n$ wiggles more an more near the vertical circle $(0, S^1)$ as $n\to \infty$). The problem with this example, is that the topology on the union of $S^1$ and $R^1$ that we should take (I guess) is the toplogy induced from $T^2$. And for this induced topology, I guess the projection to the horisontal cicle is non nullhomotopic... 

