#Volontariat bei @HCM_Bonn zu vergeben. Sofort bewerben!! #bonn #sciencejob #wisskomm #fwk15
Prof. Dr. Don Bernard Zagier entwickelt eine Vielfach-Zetafunktion.
Die werte Kollegin Dr. Astrid Slizewski vom Exzellenzcluster Hausdorff Center for Mathematics (HCM), an der Universität Bonn, sucht PR-Nachwuchs. Auf Nachfrage wurde mir versichert, dass man sich auch ohne exzessive Lust an Formeln o. ä. bewerben kann, bzw. sollte.
Zu den Rahmenbedingungen: Zweijähriges Volontariat, bei Bezahlung…
part 3 of the series on metric space topology and a brief brushing on generalised topology (toc)
Okay, so far we've basically stuck with the assumption of real numbers and the euclidean metric, absolute value. Let's branch out a bit today and look at some different possible spaces, various metrics, and more generalised topology. Get out your party drinks and put that lampshade on your head.
First, let's think about what R might look like with a different metric. Remember: to define a function as a metric, it needs to take two elements from the space and send them to a number in R, such that d(x, x) = 0, d(y, x) = d(x, y) > 0 if x ≠ y, and the triangle inequality holds. Here, I'll propose a rather simple metric: d(x, y) = 1 unless x = y. (Remember by definition that the distance from x to x has to be 0). Basically, this metric is like a computer bit: are x and y the same? Then 0. Otherwise, 1.
Check that it's a metric first: d(x, x) = 0; d(x, y) = d(y, x) = 1 > 0 if x ≠ y; and d(x, z) = 0 or 1 ≤ d(x, y) + d(x, z) = 1 or 2. So we're in business. What does an interior point look like now? Well...all points are now interior points. If we pick r = 0.5, the N(p, r) is just the point p itself, and so N(p, 0.5) is contained in X. So all discrete metric spaces are open. What about limit points? Well...there are no limit points. Because likewise, I can pick r = 0.5 and the only point that I'll find in that neighbourhood is the point itself. So all spaces with discrete metrics have no limit points, so they contain their limit points vacuously. Did I hear right? Yes: all discrete metric spaces are clopen. These spaces are basically the second simplest clopen spaces, following the ones we already covered.
Now let's try a space whose elements are not numbers. What about the space of all bounded functions defined over R? Woah, hold up. Spaces of functions? YES, FUTURE ME. This question was on your final exam! Here's the metric that your professor gave you: d(f, g) = sup|f(x) - g(x)| over all x in the domain of the functions; in this case, R. Notice we specified that the functions are all bounded, so the supremum will always exist as a real number. Here's the proof you gave on the final showing that this function is indeed a metric:
Easy first. d(f, f) = sup{f(x) - f(x)| = sup|0| = 0 identically. If f ≠ g, we can find x so that f(x) ≠ g(x). Then, |f(x) - g(x)| > 0 so the supremum over all x can't be 0.
Obviously d(f, g) = d(g, f) by nature of absolute value.
Triangle inequality: since R is closed, there exists r where sup|f(x) - h(x)| = |f(r) - h(r)|. Absolute value satisfies the triangle inequality, so |f(r) - h(r)| ≤ |f(r) - g(r)| + |g(r) - h(r)| ≤ sup|f(x) - g(x)| + sup|g(x) - h(x)| by nature of supremum. So triangle inequality holds.
At this point, I'd like to take a step back and say: "woah, we can measure the distance between functions." We can basically think of each function as a point. That's so wild. The next part of this question on the exam had to do with convergence of cauchy sequences in this metric space of functions. I haven't really talked about what it means for a sequence of functions to converge (but I do in my definition dump post!), so yall can skip this next sentence if you want... (BEING CAUCHY UNDER THIS METRIC GUARANTEES UNIFORM CONVERGENCE OMG) ...anyways.
One last cool thing: we've been talking about metric space topology...but what's a general topology? To answer this question, let me introduce another pretty important theorem we covered concerning open and closed sets:
If we have a finite collection of open sets, then their intersection is open.
For any collection of open sets (even infinite), the union is open.
If we have a finite collection of closed sets, then their union is closed.
For any collection of closed sets (even infinite), the intersection is closed.
Cool. So now: a set can come endowed with a topology (just like how a set with a metric is a metric space). It's like a two-for-one deal! A topology is just a collection of subsets in the set that follows these three rules:
the empty set and the entire set itself must be included in the collection
the intersection of any finite number of subsets from the collection must still be in the collection
the union of any number (even infinite) of subsets from the collection must still be in the collection
Sneaky eagle-eyes might have noticed that if we have a metric space, then the metric induces a topology on the set: specifically, the topology is the collection of all open sets specified by the metric. But it is possible to specify topologies without dependence on a metric. The simplest example is the coarse topology, consisting only of the empty set and the set itself. This collection fulfills the above three requirements (check if you want!) so it passes the test.
Why do I mention this topology? Well, future me, you also learned about a property called the Hausdorff property, which you can assign to sets that come with a topology. Hausdorff (aka T2) means that if you pick any two points in your set, x and y, you can find two disjoint subsets from your topology such that x is in on subset and y is in the other. (Note all metric spaces are hausdorff because you just pick N(x, r) and N(y, s) where r + s < |y - x|.) You can verify that the coarse topology applied to any set will not create a Hausdorff space.
I kind of feel like all the best theorems and properties are named after angry-sounding German surnames.
