Winding Number, An Introduction

Curves, Closed and Bounded

A path, or curve, is the continuous image of the unit interval [0,1] in real space. A closed curve starts and ends at the same point. In other words, a closed curve is the continuous image of the unit circle. This is also called a loop.

Refering to this as a closed curve is somewhat unfortunate, since every curve is topologically closed, i.e. a closed set in Rⁿ. Try to set this confusion aside, and just think of a "closed curve" as a loop.

So why is every curve topologically closed? Because the unit interval is compact, whence the curve becomes the continuous image of a compact set into a hausdorff space. The continuous image of a compact set is compact, and since the curve lives in a hausdorff space, it is closed. This generalizes to any metric space, since a metric space is always hausdorff.

Create a function f that measures the distance from every point on the curve to the origin. This is valid in any metric space. Combine this with the curve itself to build a continuous function g from [0,1] into the nonnegative reals. This function attains its maximum, hence the curve is bounded.

In summary, a curve, or loop, in a metric space, is compact, closed, and bounded.

Approaching Itself

Let g(x,y) be the distance between c(x) and c(y), where c is the curve in our metric space. This is a continuous function from the unit square into the nonnegative reals, and it happens to be 0 along the main diagonal. Assume, for a fixed x, that g approaches 0, even when y is bounded away from x, i.e. off the main diagonal. In other words, the curve approaches itself. There is a cluster point t, and a sequence y_n approaching t, such that g(x,y_n) approaches 0. By continuity, g(x,t) = 0. In other words, the curve touches itself at c(x) = c(t). Therefore, the curve cannot approach itself without touching itself.

A similar argument shows the curve cannot approach a point u that is not on the curve. The distance from c(x) to u must attain its minimum, in this case 0, whence c(x) = u. This is just another way of saying the complement of the curve is an open set.

Simple Closed Curve

A simple closed curve is a loop that does not touch itself. In other words, c is injective. Never does c(x) = c(y), unless x = y.

A more general definition says a simple closed curve is a loop that becomes injective after an appropriate reparameterization. For instance, the function c(x) might map the first half of the unit circle to 0, then map the rest of the unit circle around a loop that does not touch itself. Reparameterize c, so that only the base point maps to 0, and the rest of the circle maps onto the loop, and find an injective function that satisfies the earlier definition. So a simple closed curve is a loop that can be implemented via an injective continuous map. In general, we will assume such a reparameterization has already taken place, whence a simple closed curve is an injective loop.

Remember that in this context, continuous implies bicontinuous, hence our simple closed curve is homeomorphic to the circle. If you're a tiny ant stuck inside the curve, and you crawl along it, all the way back to start, you can't really tell you're not inside a circle.

These results extend to spheres, and other compact sets in real space, that are mapped (perhaps injectively) into a metric space.

Jordan Curves and Surfaces

A jordan arc is an injective path, and a jordan curve is an injective loop. By analogy, a jordan surface is the injective image of a sphere, floating in a metric space.

We are making our way towards the jordan curve theorem, a fact of geometry that is so obvious and intuitive, you almost don't think it requires a proof. Start at the origin and draw a curve in the plane back to start. If the curve does not cross itself, or touch itself, it cuts the plane into precisely two pieces, the inside and the outside. The curve can wiggle all about, and wander far from home, but when it finally returns to start, the fence is complete, and the dog cannot escape. It's a difficult proof, and it's not even true unless you state some of your assumptions (e.g. what is a smooth curve) very clearly. The analogous theorems in higher dimensions are also true, and also difficult to prove. A jordan surface, like the cell wall, cuts 3 dimensional space into two pieces, the inside and the outside.