Suppose (Ln) is a sequence of loops in a torus S1 × S1 converging in the Hausdorff metric to some set L in the torus. Suppose also that for each loop Ln the projection map p:S1 × S1 -> S1 defined by p(x,y) =x when restricted to Ln is not null-homotopic, 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 → S1 is not nullhomotopic.
If p : L → S1 was nullhomotopic, then it would factor through the universal cover ℝ of S1.
Let C0 ⊂ ℝ × S1 be a loop in that
separates L(0) from L(1).
The projection C ⊂ S1 × S1 of C0 is then a loop in the complement of L,
Since L lies in the complement of C and Ln → L in the Hausdorff metric,
EDITED. The following example should not be conisdered as a contre-example, 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 one-to 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:
The point is that such $L$ can be Hausodrff approximated by a sequence of circles $s_n$ that are not null-homotopic ($s_n$ 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 null-homotopic...