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One way to study a general class of mathematical objects is to embed them into a more structured class of mathematical objects; for instance, one could study manifolds by embedding them into Euclidean spaces. In these (optional) notes we study two (related) embedding theorems for topological spaces:
- The Stone-Čech compactification, which embeds locally compact Hausdorff spaces into compact Hausdorff spaces in a “universal” fashion; and
- The Urysohn metrization theorem, that shows that every second-countable normal Hausdorff space is metrizable.
We now begin the study of recurrence in topological dynamical systems – how often a non-empty open set U in X returns to intersect itself, or how often a point x in X returns to be close to itself. Not every set or point needs to return to itself; consider for instance what happens to the shift on the compactified integers . Nevertheless, we can always show that at least one set (from any open cover) returns to itself:
Theorem 1. (Simple recurrence in open covers) Let be a topological dynamical system, and let be an open cover of X. Then there exists an open set in this cover such that for infinitely many n.
Proof. By compactness of X, we can refine the open cover to a finite subcover. Now consider an orbit of some arbitrarily chosen point . By the infinite pigeonhole principle, one of the sets must contain an infinite number of the points counting multiplicity; in other words, the recurrence set is infinite. Letting be an arbitrary element of S, we thus conclude that contains for every , and the claim follows.
Exercise 1. Conversely, use Theorem 1 to deduce the infinite pigeonhole principle (i.e. that whenever is coloured into finitely many colours, one of the colour classes is infinite). Hint: look at the orbit closure of c inside , where A is the set of colours and is the colouring function.)
Now we turn from recurrence of sets to recurrence of individual points, which is a somewhat more difficult, and highlights the role of minimal dynamical systems (as introduced in the previous lecture) in the theory. We will approach the subject from two (largely equivalent) approaches, the first one being the more traditional “epsilon and delta” approach, and the second using the Stone-Čech compactification of the integers (i.e. ultrafilters).