The next theorem deals with the existence of a cycle with nice properties. Basically, you go around the compact subset of an open set with suitable contours while staying in the open set. The method involves the following simple concept.
Definition 50.7.24 A tiling of ℝ^{2} = ℂ is the union of infinitely many equally spaced vertical and horizontal lines. You can think of the small squares which result as tiles. To tile the plane or ℝ^{2} = ℂ means to consider such a union of horizontal and vertical lines. It is like graph paper. See the picture below for a representation of part of a tiling of ℂ.
Theorem 50.7.25 Let K be a compact subset of an open set, Ω. Then there exist continuous, closed, bounded variation oriented curves

while for all z

Proof: Let δ = dist
Let S denote the set of all the closed squares in this tiling which have nonempty intersection with K.Thus, all the squares of S are contained in Ω. First suppose p is a point of K which is in the interior of one of these squares in the tiling. Denote by ∂S_{k} the boundary of S_{k} one of the squares in S, oriented in the counter clockwise direction and S_{m} denote the square of S which contains the point, p in its interior. Let the edges of the square, S_{j} be

Similarly, if z
If γ_{k}^{j∗} coincides with γ_{l}^{l∗}, then the contour integrals taken over this edge are taken in opposite directions and so the edge the two squares have in common can be deleted without changing ∑ _{j,k}n
From the construction, if any of the γ_{k}^{j∗} contains a point of K then this point is on one of the four edges of S_{j} and at this point, there is at least one edge of some S_{l} which also contains this point. As just discussed, this shared edge can be deleted without changing ∑ _{k,j}n
Then as explained above, ∑ _{k=1}^{m}n
Each orientation on an edge corresponds to a direction of motion over that edge. Call such a motion over the edge a route. Initially, every vertex, (corner of a square in S) has the property there are the same number of routes to and from that vertex. When an open edge whose closure contains a point of K is deleted, every vertex either remains unchanged as to the number of routes to and from that vertex or it loses both a route away and a route to. Thus the property of having the same number of routes to and from each vertex is preserved by deleting these open edges. The isolated points which result lose all routes to and from. It follows that upon removing the isolated points you can begin at any of the remaining vertices and follow the routes leading out from this and successive vertices according to orientation and eventually return to that end. Otherwise, there would be a vertex which would have only one route leading to it which does not happen. Now if you have used all the routes out of this vertex, pick another vertex and do the same process. Otherwise, pick an unused route out of the vertex and follow it to return. Continue this way till all routes are used exactly once, resulting in closed oriented curves, Γ_{k}. Then

In case p ∈ K is on some line of the tiling, it is not on any of the Γ_{k} because Γ_{k}^{∗}∩ K = ∅ and so the continuity of z → n